Primitive. The name Eps_i is a term of type (set → prop) → set.
Axiom. (Eps_i_ax) We take the following as an axiom:
∀P : set → prop, ∀x : set, P x → P (Eps_i P)
In Proofgold the corresponding term root is c3f0de... and proposition id is 756d6e...
Definition. We define True to be ∀p : prop, p → p of type prop.
In Proofgold the corresponding term root is f81b39... and object id is 94e6a7...
Definition. We define False to be ∀p : prop, p of type prop.
In Proofgold the corresponding term root is 1db705... and object id is 266ad5...
Definition. We define not to be λA : prop ⇒ A → False of type prop → prop.
In Proofgold the corresponding term root is f30435... and object id is 0dc3c7...
Notation. We use ¬ as a prefix operator with priority 700 corresponding to applying term not.
Definition. We define and to be λA B : prop ⇒ ∀p : prop, (A → B → p) → p of type prop → prop → prop.
In Proofgold the corresponding term root is 2ba7d0... and object id is 37da83...
Notation. We use ∧ as an infix operator with priority 780 and which associates to the left corresponding to applying term and.
Definition. We define or to be λA B : prop ⇒ ∀p : prop, (A → p) → (B → p) → p of type prop → prop → prop.
In Proofgold the corresponding term root is 957746... and object id is 86ae64...
Notation. We use ∨ as an infix operator with priority 785 and which associates to the left corresponding to applying term or.
Definition. We define iff to be λA B : prop ⇒ and (A → B) (B → A) of type prop → prop → prop.
In Proofgold the corresponding term root is 98aaaf... and object id is 6588e5...
Notation. We use ↔ as an infix operator with priority 805 and no associativity corresponding to applying term iff.
Beginning of Section Eq
Variable A : SType
Definition. We define eq to be λx y : A ⇒ ∀Q : A → A → prop, Q x y → Q y x of type A → A → prop.
Definition. We define neq to be λx y : A ⇒ ¬ eq x y of type A → A → prop.
End of Section Eq
Notation. We use = as an infix operator with priority 502 and no associativity corresponding to applying term eq.
Notation. We use ≠ as an infix operator with priority 502 and no associativity corresponding to applying term neq.
Beginning of Section FE
Variable A B : SType
Axiom. (func_ext) We take the following as an axiom:
∀f g : A → B, (∀x : A, f x = g x) → f = g
End of Section FE
Beginning of Section Ex
Variable A : SType
Definition. We define ex to be λQ : A → prop ⇒ ∀P : prop, (∀x : A, Q x → P) → P of type (A → prop) → prop.
End of Section Ex
Notation. We use ∃ x...y [possibly with ascriptions] , B as a binder notation corresponding to a term constructed using ex.
Axiom. (prop_ext) We take the following as an axiom:
∀p q : prop, iff p q → p = q
In Proofgold the corresponding term root is 00ac81... and proposition id is 1d1e56...
Primitive. The name In is a term of type set → set → prop.
Notation. We use ∈ as an infix operator with priority 500 and no associativity corresponding to applying term In. Furthermore, we may write ∀ x ∈ A, B to mean ∀ x : set, x ∈ A → B.
Definition. We define Subq to be λA B ⇒ ∀x ∈ A, x ∈ B of type set → set → prop.
In Proofgold the corresponding term root is 81c0ef... and object id is 317007...
Notation. We use ⊆ as an infix operator with priority 500 and no associativity corresponding to applying term Subq. Furthermore, we may write ∀ x ⊆ A, B to mean ∀ x : set, x ⊆ A → B.
Axiom. (set_ext) We take the following as an axiom:
∀X Y : set, X ⊆ Y → Y ⊆ X → X = Y
In Proofgold the corresponding term root is b3b698... and proposition id is 21238a...
Axiom. (In_ind) We take the following as an axiom:
∀P : set → prop, (∀X : set, (∀x ∈ X, P x) → P X) → ∀X : set, P X
In Proofgold the corresponding term root is 045211... and proposition id is 20faa5...
Notation. We use ∃ x...y [possibly with ascriptions] , B as a binder notation corresponding to a term constructed using ex and handling ∈ or ⊆ ascriptions using and.
Primitive. The name Empty is a term of type set.
Axiom. (EmptyAx) We take the following as an axiom:
¬ ∃x : set, x ∈ Empty
In Proofgold the corresponding term root is 4da931... and proposition id is c7b0d7...
Primitive. The name ⋃ is a term of type set → set.
Axiom. (UnionEq) We take the following as an axiom:
∀X x, x ∈ ⋃ X ↔ ∃Y, x ∈ Y ∧ Y ∈ X
In Proofgold the corresponding term root is bf4c26... and proposition id is 40660f...
Primitive. The name 𝒫 is a term of type set → set.
Axiom. (PowerEq) We take the following as an axiom:
∀X Y : set, Y ∈ 𝒫 X ↔ Y ⊆ X
In Proofgold the corresponding term root is 746e5a... and proposition id is d67d71...
Primitive. The name Repl is a term of type set → (set → set) → set.
Notation. {B| x ∈ A} is notation for Repl A (λ x . B).
Axiom. (ReplEq) We take the following as an axiom:
∀A : set, ∀F : set → set, ∀y : set, y ∈ {F x|x ∈ A} ↔ ∃x ∈ A, y = F x
In Proofgold the corresponding term root is 4f2357... and proposition id is 471122...
Definition. We define TransSet to be λU : set ⇒ ∀x ∈ U, x ⊆ U of type set → prop.
In Proofgold the corresponding term root is 538bb7... and object id is e69f92...
Definition. We define Union_closed to be λU : set ⇒ ∀X : set, X ∈ U → ⋃ X ∈ U of type set → prop.
In Proofgold the corresponding term root is 57561d... and object id is e45ed9...
Definition. We define Power_closed to be λU : set ⇒ ∀X : set, X ∈ U → 𝒫 X ∈ U of type set → prop.
In Proofgold the corresponding term root is 8b9cc0... and object id is d2954e...
Definition. We define Repl_closed to be λU : set ⇒ ∀X : set, X ∈ U → ∀F : set → set, (∀x : set, x ∈ X → F x ∈ U) → {F x|x ∈ X} ∈ U of type set → prop.
In Proofgold the corresponding term root is 5574bb... and object id is 513534...
Definition. We define ZF_closed to be λU : set ⇒ Union_closed U ∧ Power_closed U ∧ Repl_closed U of type set → prop.
In Proofgold the corresponding term root is 1bd4aa... and object id is e1eccf...
Primitive. The name UnivOf is a term of type set → set.
Axiom. (UnivOf_In) We take the following as an axiom:
∀N : set, N ∈ UnivOf N
In Proofgold the corresponding term root is efcc16... and proposition id is e805d4...
Axiom. (UnivOf_TransSet) We take the following as an axiom:
∀N : set, TransSet (UnivOf N)
In Proofgold the corresponding term root is 053575... and proposition id is cf4f5c...
Axiom. (UnivOf_ZF_closed) We take the following as an axiom:
∀N : set, ZF_closed (UnivOf N)
In Proofgold the corresponding term root is 770e35... and proposition id is 0e7363...
Axiom. (UnivOf_Min) We take the following as an axiom:
∀N U : set, N ∈ U → TransSet U → ZF_closed U → UnivOf N ⊆ U
In Proofgold the corresponding term root is 042dda... and proposition id is 1f7933...
Theorem. (andI)
∀A B : prop, A → B → A ∧ B
In Proofgold the corresponding term root is 3c4f59... and proposition id is 3bf198...
Proof:
An exact proof term for the current goal is (λA B a b P H ⇒ H a b).
∎
Theorem. (orIL)
∀A B : prop, A → A ∨ B
In Proofgold the corresponding term root is 55be8c... and proposition id is 382673...
Proof:
An exact proof term for the current goal is (λA B a P H1 H2 ⇒ H1 a).
∎
Theorem. (orIR)
∀A B : prop, B → A ∨ B
In Proofgold the corresponding term root is 3c63fc... and proposition id is 8d8e40...
Proof:
An exact proof term for the current goal is (λA B b P H1 H2 ⇒ H2 b).
∎
Theorem. (iffI)
∀A B : prop, (A → B) → (B → A) → (A ↔ B)
In Proofgold the corresponding term root is 7516b9... and proposition id is 4d966a...
Proof:
An exact proof term for the current goal is (λA B ⇒ andI (A → B) (B → A)).
∎
Theorem. (pred_ext)
∀P Q : set → prop, (∀x, P x ↔ Q x) → P = Q
In Proofgold the corresponding term root is e9e64b... and proposition id is d4b2d3...
Proof:
Let P and Q be given.
Assume H1.
Apply func_ext set prop to the current goal.
Let x be given.
Apply prop_ext to the current goal.
We will prove P x ↔ Q x.
An exact proof term for the current goal is H1 x.
∎
Definition. We define nIn to be λx X ⇒ ¬ In x X of type set → set → prop.
In Proofgold the corresponding term root is 36808c... and object id is 253451...
Notation. We use ∉ as an infix operator with priority 502 and no associativity corresponding to applying term nIn.
Theorem. (EmptyE)
∀x : set, x ∉ Empty
In Proofgold the corresponding term root is 93a77a... and proposition id is 8349ab...
Proof:
Let x be given.
Assume H.
Apply EmptyAx to the current goal.
We use x to witness the existential quantifier.
An exact proof term for the current goal is H.
∎
Theorem. (PowerI)
∀X Y : set, Y ⊆ X → Y ∈ 𝒫 X
In Proofgold the corresponding term root is 7143d7... and proposition id is 3c48d3...
Proof:
Let X and Y be given.
Apply PowerEq X Y to the current goal.
An exact proof term for the current goal is (λ_ H ⇒ H).
∎
Theorem. (Subq_Empty)
∀X : set, Empty ⊆ X
In Proofgold the corresponding term root is a517cd... and proposition id is 2faec3...
Proof:
An exact proof term for the current goal is (λ(X x : set)(H : x ∈ Empty) ⇒ EmptyE x H (x ∈ X)).
∎
Theorem. (Empty_In_Power)
∀X : set, Empty ∈ 𝒫 X
In Proofgold the corresponding term root is d2656b... and proposition id is 487855...
Proof:
An exact proof term for the current goal is (λX : set ⇒ PowerI X Empty (Subq_Empty X)).
∎
Theorem. (xm)
∀P : prop, P ∨ ¬ P
In Proofgold the corresponding term root is 62e4c3... and proposition id is 7b6737...
Proof:
Let P of type prop be given.
Set p1 to be the term λx : set ⇒ x = Empty ∨ P.
Set p2 to be the term λx : set ⇒ x ≠ Empty ∨ P.
We prove the intermediate claim L1: p1 Empty.
We will prove (Empty = Empty ∨ P).
Apply orIL to the current goal.
An exact proof term for the current goal is (λq H ⇒ H).
We prove the intermediate claim L2: (Eps_i p1) = Empty ∨ P.
An exact proof term for the current goal is (Eps_i_ax p1 Empty L1).
We prove the intermediate claim L3: p2 (𝒫 Empty).
We will prove ¬ (𝒫 Empty = Empty) ∨ P.
Apply orIL to the current goal.
Assume H1: 𝒫 Empty = Empty.
Apply EmptyE Empty to the current goal.
We will prove Empty ∈ Empty.
rewrite the current goal using H1 (from right to left) at position 2.
Apply Empty_In_Power to the current goal.
We prove the intermediate claim L4: Eps_i p2 ≠ Empty ∨ P.
An exact proof term for the current goal is (Eps_i_ax p2 (𝒫 Empty) L3).
Apply L2 to the current goal.
Assume H1: Eps_i p1 = Empty.
Apply L4 to the current goal.
Assume H2: Eps_i p2 ≠ Empty.
We will prove P ∨ ¬ P.
Apply orIR to the current goal.
We will prove ¬ P.
Assume H3: P.
We prove the intermediate claim L5: p1 = p2.
Apply pred_ext to the current goal.
Let x be given.
Apply iffI to the current goal.
Assume H4.
We will prove (¬ (x = Empty) ∨ P).
Apply orIR to the current goal.
We will prove P.
An exact proof term for the current goal is H3.
Assume H4.
We will prove (x = Empty ∨ P).
Apply orIR to the current goal.
We will prove P.
An exact proof term for the current goal is H3.
Apply H2 to the current goal.
rewrite the current goal using L5 (from right to left).
An exact proof term for the current goal is H1.
Assume H2: P.
We will prove P ∨ ¬ P.
Apply orIL to the current goal.
We will prove P.
An exact proof term for the current goal is H2.
Assume H1: P.
We will prove P ∨ ¬ P.
Apply orIL to the current goal.
We will prove P.
An exact proof term for the current goal is H1.
∎
Theorem. (FalseE)
False → ∀p : prop, p
In Proofgold the corresponding term root is 0dcfe3... and proposition id is f9a230...
Proof:
An exact proof term for the current goal is (λH ⇒ H).
∎
Theorem. (andEL)
∀A B : prop, A ∧ B → A
In Proofgold the corresponding term root is c2f006... and proposition id is 718be4...
Proof:
An exact proof term for the current goal is (λA B H ⇒ H A (λa b ⇒ a)).
∎
Theorem. (andER)
∀A B : prop, A ∧ B → B
In Proofgold the corresponding term root is f66b82... and proposition id is 5caea9...
Proof:
An exact proof term for the current goal is (λA B H ⇒ H B (λa b ⇒ b)).
∎
Beginning of Section PropN
Variable P1 P2 P3 : prop
Theorem. (and3I)
P1 → P2 → P3 → P1 ∧ P2 ∧ P3
In Proofgold the corresponding term root is 23ac9c... and proposition id is cb999b...
Proof:
An exact proof term for the current goal is (λH1 H2 H3 ⇒ andI (P1 ∧ P2) P3 (andI P1 P2 H1 H2) H3).
∎
Theorem. (and3E)
P1 ∧ P2 ∧ P3 → (∀p : prop, (P1 → P2 → P3 → p) → p)
In Proofgold the corresponding term root is 445f76... and proposition id is 50fabe...
Proof:
An exact proof term for the current goal is (λu p H ⇒ u p (λu u3 ⇒ u p (λu1 u2 ⇒ H u1 u2 u3))).
∎
Theorem. (or3I1)
P1 → P1 ∨ P2 ∨ P3
In Proofgold the corresponding term root is e44f34... and proposition id is ef9071...
Proof:
An exact proof term for the current goal is (λu ⇒ orIL (P1 ∨ P2) P3 (orIL P1 P2 u)).
∎
Theorem. (or3I2)
P2 → P1 ∨ P2 ∨ P3
In Proofgold the corresponding term root is 8df6e1... and proposition id is 827a91...
Proof:
An exact proof term for the current goal is (λu ⇒ orIL (P1 ∨ P2) P3 (orIR P1 P2 u)).
∎
Theorem. (or3I3)
P3 → P1 ∨ P2 ∨ P3
In Proofgold the corresponding term root is 017a00... and proposition id is 3b7f5a...
Proof:
An exact proof term for the current goal is (orIR (P1 ∨ P2) P3).
∎
Theorem. (or3E)
P1 ∨ P2 ∨ P3 → (∀p : prop, (P1 → p) → (P2 → p) → (P3 → p) → p)
In Proofgold the corresponding term root is 5d069d... and proposition id is 2120c9...
Proof:
An exact proof term for the current goal is (λu p H1 H2 H3 ⇒ u p (λu ⇒ u p H1 H2) H3).
∎
Variable P4 : prop
Theorem. (and4I)
P1 → P2 → P3 → P4 → P1 ∧ P2 ∧ P3 ∧ P4
In Proofgold the corresponding term root is ed32cb... and proposition id is 906aaf...
Proof:
An exact proof term for the current goal is (λH1 H2 H3 H4 ⇒ andI (P1 ∧ P2 ∧ P3) P4 (and3I H1 H2 H3) H4).
∎
Variable P5 : prop
Theorem. (and5I)
P1 → P2 → P3 → P4 → P5 → P1 ∧ P2 ∧ P3 ∧ P4 ∧ P5
In Proofgold the corresponding term root is 041ced... and proposition id is 11fc47...
Proof:
An exact proof term for the current goal is (λH1 H2 H3 H4 H5 ⇒ andI (P1 ∧ P2 ∧ P3 ∧ P4) P5 (and4I H1 H2 H3 H4) H5).
∎
Variable P6 : prop
Theorem. (and6I)
P1 → P2 → P3 → P4 → P5 → P6 → P1 ∧ P2 ∧ P3 ∧ P4 ∧ P5 ∧ P6
In Proofgold the corresponding term root is 9817c7... and proposition id is a5dc29...
Proof:
An exact proof term for the current goal is (λH1 H2 H3 H4 H5 H6 ⇒ andI (P1 ∧ P2 ∧ P3 ∧ P4 ∧ P5) P6 (and5I H1 H2 H3 H4 H5) H6).
∎
Variable P7 : prop
Theorem. (and7I)
P1 → P2 → P3 → P4 → P5 → P6 → P7 → P1 ∧ P2 ∧ P3 ∧ P4 ∧ P5 ∧ P6 ∧ P7
In Proofgold the corresponding term root is 6a4d8c... and proposition id is c89388...
Proof:
An exact proof term for the current goal is (λH1 H2 H3 H4 H5 H6 H7 ⇒ andI (P1 ∧ P2 ∧ P3 ∧ P4 ∧ P5 ∧ P6) P7 (and6I H1 H2 H3 H4 H5 H6) H7).
∎
End of Section PropN
Theorem. (not_or_and_demorgan)
∀A B : prop, ¬ (A ∨ B) → ¬ A ∧ ¬ B
In Proofgold the corresponding term root is 8733b0... and proposition id is 27db91...
Proof:
Let A and B be given.
Assume u: ¬ (A ∨ B).
Apply andI to the current goal.
We will prove ¬ A.
Assume a: A.
An exact proof term for the current goal is (u (orIL A B a)).
We will prove ¬ B.
Assume b: B.
An exact proof term for the current goal is (u (orIR A B b)).
∎
Theorem. (not_ex_all_demorgan_i)
∀P : set → prop, (¬ ∃x, P x) → ∀x, ¬ P x
In Proofgold the corresponding term root is 73523e... and proposition id is 075ddf...
Proof:
Let P be given.
Assume H1.
Let x be given.
Assume H2.
Apply H1 to the current goal.
We use x to witness the existential quantifier.
An exact proof term for the current goal is H2.
∎
Theorem. (iffEL)
∀A B : prop, (A ↔ B) → A → B
In Proofgold the corresponding term root is 0e94c6... and proposition id is 9a07f6...
Proof:
An exact proof term for the current goal is (λA B ⇒ andEL (A → B) (B → A)).
∎
Theorem. (iffER)
∀A B : prop, (A ↔ B) → B → A
In Proofgold the corresponding term root is 54702a... and proposition id is f5dfc0...
Proof:
An exact proof term for the current goal is (λA B ⇒ andER (A → B) (B → A)).
∎
Theorem. (iff_refl)
∀A : prop, A ↔ A
In Proofgold the corresponding term root is 01338d... and proposition id is 7b71b5...
Proof:
An exact proof term for the current goal is (λA : prop ⇒ andI (A → A) (A → A) (λH : A ⇒ H) (λH : A ⇒ H)).
∎
Theorem. (iff_sym)
∀A B : prop, (A ↔ B) → (B ↔ A)
In Proofgold the corresponding term root is 00bac1... and proposition id is 11f51b...
Proof:
Let A and B be given.
Assume H1: (A → B) ∧ (B → A).
Apply H1 to the current goal.
Assume H2: A → B.
Assume H3: B → A.
An exact proof term for the current goal is iffI B A H3 H2.
∎
Theorem. (iff_trans)
∀A B C : prop, (A ↔ B) → (B ↔ C) → (A ↔ C)
In Proofgold the corresponding term root is 363285... and proposition id is c3b8f1...
Proof:
Let A, B and C be given.
Assume H1: A ↔ B.
Assume H2: B ↔ C.
Apply H1 to the current goal.
Assume H3: A → B.
Assume H4: B → A.
Apply H2 to the current goal.
Assume H5: B → C.
Assume H6: C → B.
An exact proof term for the current goal is (iffI A C (λH ⇒ H5 (H3 H)) (λH ⇒ H4 (H6 H))).
∎
Theorem. (eq_i_tra)
∀x y z, x = y → y = z → x = z
In Proofgold the corresponding term root is 193bbf... and proposition id is 7ecbc4...
Proof:
Let x, y and z be given.
Assume H1 H2.
rewrite the current goal using H2 (from right to left).
An exact proof term for the current goal is H1.
∎
Theorem. (neq_i_sym)
∀x y, x ≠ y → y ≠ x
In Proofgold the corresponding term root is 5c0ec9... and proposition id is 1a9898...
Proof:
Let x and y be given.
Assume H1 H2.
Apply H1 to the current goal.
Use symmetry.
An exact proof term for the current goal is H2.
∎
Theorem. (Eps_i_ex)
∀P : set → prop, (∃x, P x) → P (Eps_i P)
In Proofgold the corresponding term root is be36f0... and proposition id is 6a2832...
Proof:
Let P be given.
Assume H1.
Apply H1 to the current goal.
Let x be given.
Assume H2.
An exact proof term for the current goal is Eps_i_ax P x H2.
∎
Theorem. (prop_ext_2)
∀p q : prop, (p → q) → (q → p) → p = q
In Proofgold the corresponding term root is a9ede2... and proposition id is a56fc7...
Proof:
Let p and q be given.
Assume H1 H2.
Apply prop_ext to the current goal.
Apply iffI to the current goal.
An exact proof term for the current goal is H1.
An exact proof term for the current goal is H2.
∎
Theorem. (Subq_ref)
∀X : set, X ⊆ X
In Proofgold the corresponding term root is 4c5395... and proposition id is 2efbac...
Proof:
An exact proof term for the current goal is (λ(X x : set)(H : x ∈ X) ⇒ H).
∎
Theorem. (Subq_tra)
∀X Y Z : set, X ⊆ Y → Y ⊆ Z → X ⊆ Z
In Proofgold the corresponding term root is 79accb... and proposition id is fed662...
Proof:
An exact proof term for the current goal is (λ(X Y Z : set)(H1 : X ⊆ Y)(H2 : Y ⊆ Z)(x : set)(H : x ∈ X) ⇒ (H2 x (H1 x H))).
∎
Theorem. (Empty_Subq_eq)
∀X : set, X ⊆ Empty → X = Empty
In Proofgold the corresponding term root is 2dd959... and proposition id is 988691...
Proof:
Let X be given.
Assume H1: X ⊆ Empty.
Apply set_ext to the current goal.
An exact proof term for the current goal is H1.
An exact proof term for the current goal is (Subq_Empty X).
∎
Theorem. (Empty_eq)
∀X : set, (∀x, x ∉ X) → X = Empty
In Proofgold the corresponding term root is 3de0fd... and proposition id is 8b915b...
Proof:
Let X be given.
Assume H1: ∀x, x ∉ X.
Apply Empty_Subq_eq to the current goal.
Let x be given.
Assume H2: x ∈ X.
We will prove False.
An exact proof term for the current goal is (H1 x H2).
∎
Theorem. (UnionI)
∀X x Y : set, x ∈ Y → Y ∈ X → x ∈ ⋃ X
In Proofgold the corresponding term root is 51b549... and proposition id is bade05...
Proof:
Let X, x and Y be given.
Assume H1: x ∈ Y.
Assume H2: Y ∈ X.
Apply UnionEq X x to the current goal.
Assume _ H3.
Apply H3 to the current goal.
We will prove ∃Y : set, x ∈ Y ∧ Y ∈ X.
We use Y to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is H1.
An exact proof term for the current goal is H2.
∎
Theorem. (UnionE)
∀X x : set, x ∈ ⋃ X → ∃Y : set, x ∈ Y ∧ Y ∈ X
In Proofgold the corresponding term root is db4bc1... and proposition id is ab781b...
Proof:
An exact proof term for the current goal is (λX x : set ⇒ iffEL (x ∈ ⋃ X) (∃Y : set, x ∈ Y ∧ Y ∈ X) (UnionEq X x)).
∎
Theorem. (UnionE_impred)
∀X x : set, x ∈ ⋃ X → ∀p : prop, (∀Y : set, x ∈ Y → Y ∈ X → p) → p
In Proofgold the corresponding term root is 8efd57... and proposition id is d621b2...
Proof:
Let X and x be given.
Assume H1.
Let p be given.
Assume Hp.
Apply UnionE X x H1 to the current goal.
Let x be given.
Assume H2.
Apply H2 to the current goal.
An exact proof term for the current goal is Hp x.
∎
Theorem. (PowerE)
∀X Y : set, Y ∈ 𝒫 X → Y ⊆ X
In Proofgold the corresponding term root is 149030... and proposition id is 647070...
Proof:
Let X and Y be given.
Apply PowerEq X Y to the current goal.
An exact proof term for the current goal is (λH _ ⇒ H).
∎
Theorem. (Self_In_Power)
∀X : set, X ∈ 𝒫 X
In Proofgold the corresponding term root is f81ef2... and proposition id is 8336ff...
Proof:
An exact proof term for the current goal is (λX : set ⇒ PowerI X X (Subq_ref X)).
∎
Theorem. (dneg)
∀P : prop, ¬ ¬ P → P
In Proofgold the corresponding term root is 852f34... and proposition id is b7cc63...
Proof:
Let P be given.
Assume H1.
Apply xm P to the current goal.
An exact proof term for the current goal is (λH ⇒ H).
Assume H2: ¬ P.
We will prove False.
An exact proof term for the current goal is H1 H2.
∎
Theorem. (not_all_ex_demorgan_i)
∀P : set → prop, ¬ (∀x, P x) → ∃x, ¬ P x
In Proofgold the corresponding term root is 9bfa06... and proposition id is 4b1474...
Proof:
Let P be given.
Assume u: ¬ ∀x, P x.
Apply dneg to the current goal.
Assume v: ¬ ∃x, ¬ P x.
Apply u to the current goal.
Let x be given.
Apply dneg to the current goal.
Assume w: ¬ P x.
An exact proof term for the current goal is (not_ex_all_demorgan_i (λx ⇒ ¬ P x) v x w).
∎
Theorem. (eq_or_nand)
or = (λx y : prop ⇒ ¬ (¬ x ∧ ¬ y))
In Proofgold the corresponding term root is 24ac29... and proposition id is d212ef...
Proof:
Apply func_ext prop (prop → prop) to the current goal.
Let x be given.
Apply func_ext prop prop to the current goal.
Let y be given.
Apply prop_ext_2 to the current goal.
Assume H1: x ∨ y.
Assume H2: ¬ x ∧ ¬ y.
Apply H2 to the current goal.
Assume H3 H4.
An exact proof term for the current goal is (H1 False H3 H4).
Assume H1: ¬ (¬ x ∧ ¬ y).
Apply (xm x) to the current goal.
Assume H2: x.
Apply orIL to the current goal.
An exact proof term for the current goal is H2.
Assume H2: ¬ x.
Apply (xm y) to the current goal.
Assume H3: y.
Apply orIR to the current goal.
An exact proof term for the current goal is H3.
Assume H3: ¬ y.
Apply H1 to the current goal.
An exact proof term for the current goal is (andI (¬ x) (¬ y) H2 H3).
∎
Definition. We define exactly1of2 to be λA B : prop ⇒ A ∧ ¬ B ∨ ¬ A ∧ B of type prop → prop → prop.
In Proofgold the corresponding term root is 163602... and object id is a0b944...
Theorem. (exactly1of2_I1)
∀A B : prop, A → ¬ B → exactly1of2 A B
In Proofgold the corresponding term root is 8e1e45... and proposition id is 4f94fa...
Proof:
Let A and B be given.
Assume HA: A.
Assume HB: ¬ B.
We will prove A ∧ ¬ B ∨ ¬ A ∧ B.
Apply orIL to the current goal.
We will prove A ∧ ¬ B.
An exact proof term for the current goal is (andI A (¬ B) HA HB).
∎
Theorem. (exactly1of2_I2)
∀A B : prop, ¬ A → B → exactly1of2 A B
In Proofgold the corresponding term root is c9fd90... and proposition id is 6c1ad9...
Proof:
Let A and B be given.
Assume HA: ¬ A.
Assume HB: B.
We will prove A ∧ ¬ B ∨ ¬ A ∧ B.
Apply orIR to the current goal.
We will prove ¬ A ∧ B.
An exact proof term for the current goal is (andI (¬ A) B HA HB).
∎
Theorem. (exactly1of2_E)
∀A B : prop, exactly1of2 A B → ∀p : prop, (A → ¬ B → p) → (¬ A → B → p) → p
In Proofgold the corresponding term root is 29b901... and proposition id is c67434...
Proof:
Let A and B be given.
Assume H1: exactly1of2 A B.
Let p be given.
Assume H2: A → ¬ B → p.
Assume H3: ¬ A → B → p.
Apply (H1 p) to the current goal.
An exact proof term for the current goal is (λH4 : A ∧ ¬ B ⇒ H4 p H2).
An exact proof term for the current goal is (λH4 : ¬ A ∧ B ⇒ H4 p H3).
∎
Theorem. (exactly1of2_or)
∀A B : prop, exactly1of2 A B → A ∨ B
In Proofgold the corresponding term root is 18af73... and proposition id is e092f6...
Proof:
Let A and B be given.
Assume H1: exactly1of2 A B.
Apply (exactly1of2_E A B H1 (A ∨ B)) to the current goal.
An exact proof term for the current goal is (λ(HA : A)(_ : ¬ B) ⇒ orIL A B HA).
An exact proof term for the current goal is (λ(_ : ¬ A)(HB : B) ⇒ orIR A B HB).
∎
Theorem. (ReplI)
∀A : set, ∀F : set → set, ∀x : set, x ∈ A → F x ∈ {F x|x ∈ A}
In Proofgold the corresponding term root is b146c5... and proposition id is f84982...
Proof:
Let A, F and x be given.
Assume H1.
Apply ReplEq A F (F x) to the current goal.
Assume _ H2.
Apply H2 to the current goal.
We will prove ∃x' ∈ A, F x = F x'.
We use x to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is H1.
An exact proof term for the current goal is (λq H ⇒ H).
∎
Theorem. (ReplE)
∀A : set, ∀F : set → set, ∀y : set, y ∈ {F x|x ∈ A} → ∃x ∈ A, y = F x
In Proofgold the corresponding term root is 311cdf... and proposition id is d07e78...
Proof:
Let A, F and y be given.
Apply ReplEq A F y to the current goal.
An exact proof term for the current goal is (λH _ ⇒ H).
∎
Theorem. (ReplE_impred)
∀A : set, ∀F : set → set, ∀y : set, y ∈ {F x|x ∈ A} → ∀p : prop, (∀x : set, x ∈ A → y = F x → p) → p
In Proofgold the corresponding term root is 31e90a... and proposition id is 34fefd...
Proof:
Let A, F and y be given.
Assume H1.
Apply ReplE A F y H1 to the current goal.
Let x be given.
Assume H2.
Apply H2 to the current goal.
Assume H3 H4.
Let p be given.
Assume Hp.
An exact proof term for the current goal is Hp x H3 H4.
∎
Theorem. (ReplE')
∀X, ∀f : set → set, ∀p : set → prop, (∀x ∈ X, p (f x)) → ∀y ∈ {f x|x ∈ X}, p y
In Proofgold the corresponding term root is e374fe... and proposition id is d4b7b0...
Proof:
Let X, f and p be given.
Assume H1.
Let y be given.
Assume Hy.
Apply ReplE_impred X f y Hy to the current goal.
Let x be given.
Assume Hx: x ∈ X.
Assume Hx2: y = f x.
We will prove p y.
rewrite the current goal using Hx2 (from left to right).
An exact proof term for the current goal is H1 x Hx.
∎
Theorem. (Repl_Empty)
∀F : set → set, {F x|x ∈ Empty} = Empty
In Proofgold the corresponding term root is 2af2ac... and proposition id is 0835bc...
Proof:
Let F be given.
Apply (Empty_eq {F x|x ∈ Empty}) to the current goal.
Let y be given.
Assume H1: y ∈ {F x|x ∈ Empty}.
Apply (ReplE_impred Empty F y H1) to the current goal.
Let x be given.
Assume H2: x ∈ Empty.
Assume _.
An exact proof term for the current goal is (EmptyE x H2).
∎
Theorem. (ReplEq_ext_sub)
∀X, ∀F G : set → set, (∀x ∈ X, F x = G x) → {F x|x ∈ X} ⊆ {G x|x ∈ X}
In Proofgold the corresponding term root is eb4c1d... and proposition id is 56aad0...
Proof:
Let X, F and G be given.
Assume H1: ∀x ∈ X, F x = G x.
Let y be given.
Assume Hy: y ∈ {F x|x ∈ X}.
Apply ReplE_impred X F y Hy to the current goal.
Let x be given.
Assume Hx: x ∈ X.
Assume H2: y = F x.
We will prove y ∈ {G x|x ∈ X}.
rewrite the current goal using H2 (from left to right).
We will prove F x ∈ {G x|x ∈ X}.
rewrite the current goal using H1 x Hx (from left to right).
We will prove G x ∈ {G x|x ∈ X}.
Apply ReplI to the current goal.
An exact proof term for the current goal is Hx.
∎
Theorem. (ReplEq_ext)
∀X, ∀F G : set → set, (∀x ∈ X, F x = G x) → {F x|x ∈ X} = {G x|x ∈ X}
In Proofgold the corresponding term root is 81eb90... and proposition id is 781846...
Proof:
Let X, F and G be given.
Assume H1: ∀x ∈ X, F x = G x.
Apply set_ext to the current goal.
An exact proof term for the current goal is ReplEq_ext_sub X F G H1.
Apply ReplEq_ext_sub X G F to the current goal.
Let x be given.
Assume Hx.
Use symmetry.
An exact proof term for the current goal is H1 x Hx.
∎
Theorem. (Repl_inv_eq)
∀P : set → prop, ∀f g : set → set, (∀x, P x → g (f x) = x) → ∀X, (∀x ∈ X, P x) → {g y|y ∈ {f x|x ∈ X}} = X
In Proofgold the corresponding term root is 968c7a... and proposition id is 9ce73d...
Proof:
Let P, f and g be given.
Assume H1.
Let X be given.
Assume HX.
Apply set_ext to the current goal.
Let w be given.
Assume Hw: w ∈ {g y|y ∈ {f x|x ∈ X}}.
Apply ReplE_impred {f x|x ∈ X} g w Hw to the current goal.
Let y be given.
Assume Hy: y ∈ {f x|x ∈ X}.
Assume Hwy: w = g y.
Apply ReplE_impred X f y Hy to the current goal.
Let x be given.
Assume Hx: x ∈ X.
Assume Hyx: y = f x.
We will prove w ∈ X.
rewrite the current goal using Hwy (from left to right).
rewrite the current goal using Hyx (from left to right).
We will prove g (f x) ∈ X.
rewrite the current goal using H1 x (HX x Hx) (from left to right).
An exact proof term for the current goal is Hx.
Let x be given.
Assume Hx: x ∈ X.
rewrite the current goal using H1 x (HX x Hx) (from right to left).
We will prove g (f x) ∈ {g y|y ∈ {f x|x ∈ X}}.
Apply ReplI to the current goal.
We will prove f x ∈ {f x|x ∈ X}.
Apply ReplI to the current goal.
An exact proof term for the current goal is Hx.
∎
Theorem. (Repl_invol_eq)
∀P : set → prop, ∀f : set → set, (∀x, P x → f (f x) = x) → ∀X, (∀x ∈ X, P x) → {f y|y ∈ {f x|x ∈ X}} = X
In Proofgold the corresponding term root is e15a6b... and proposition id is 5faff0...
Proof:
Let P and f be given.
Assume H1.
An exact proof term for the current goal is Repl_inv_eq P f f H1.
∎
Definition. We define If_i to be (λp x y ⇒ Eps_i (λz : set ⇒ p ∧ z = x ∨ ¬ p ∧ z = y)) of type prop → set → set → set.
In Proofgold the corresponding term root is b8ff52... and object id is 76c4c0...
Notation. if cond then T else E is notation corresponding to If_i type cond T E where type is the inferred type of T.
Theorem. (If_i_correct)
∀p : prop, ∀x y : set, p ∧ (if p then x else y) = x ∨ ¬ p ∧ (if p then x else y) = y
In Proofgold the corresponding term root is 2a17ae... and proposition id is b5c26b...
Proof:
Let p, x and y be given.
Apply (xm p) to the current goal.
Assume H1: p.
We prove the intermediate claim L1: p ∧ x = x ∨ ¬ p ∧ x = y.
Apply orIL to the current goal.
Apply andI to the current goal.
An exact proof term for the current goal is H1.
Use reflexivity.
An exact proof term for the current goal is (Eps_i_ax (λz : set ⇒ p ∧ z = x ∨ ¬ p ∧ z = y) x L1).
Assume H1: ¬ p.
We prove the intermediate claim L1: p ∧ y = x ∨ ¬ p ∧ y = y.
Apply orIR to the current goal.
Apply andI to the current goal.
An exact proof term for the current goal is H1.
Use reflexivity.
An exact proof term for the current goal is (Eps_i_ax (λz : set ⇒ p ∧ z = x ∨ ¬ p ∧ z = y) y L1).
∎
Theorem. (If_i_0)
∀p : prop, ∀x y : set, ¬ p → (if p then x else y) = y
In Proofgold the corresponding term root is fde1a8... and proposition id is a9abd3...
Proof:
Let p, x and y be given.
Assume H1: ¬ p.
Apply (If_i_correct p x y) to the current goal.
Assume H2: p ∧ (if p then x else y) = x.
An exact proof term for the current goal is (H1 (andEL p ((if p then x else y) = x) H2) ((if p then x else y) = y)).
Assume H2: ¬ p ∧ (if p then x else y) = y.
An exact proof term for the current goal is (andER (¬ p) ((if p then x else y) = y) H2).
∎
Theorem. (If_i_1)
∀p : prop, ∀x y : set, p → (if p then x else y) = x
In Proofgold the corresponding term root is b1aa3f... and proposition id is a93c2c...
Proof:
Let p, x and y be given.
Assume H1: p.
Apply (If_i_correct p x y) to the current goal.
Assume H2: p ∧ (if p then x else y) = x.
An exact proof term for the current goal is (andER p ((if p then x else y) = x) H2).
Assume H2: ¬ p ∧ (if p then x else y) = y.
An exact proof term for the current goal is (andEL (¬ p) ((if p then x else y) = y) H2 H1 ((if p then x else y) = x)).
∎
Theorem. (If_i_or)
∀p : prop, ∀x y : set, (if p then x else y) = x ∨ (if p then x else y) = y
In Proofgold the corresponding term root is 3e1439... and proposition id is 4de742...
Proof:
Let p, x and y be given.
Apply (xm p) to the current goal.
Assume H1: p.
Apply orIL to the current goal.
An exact proof term for the current goal is (If_i_1 p x y H1).
Assume H1: ¬ p.
Apply orIR to the current goal.
An exact proof term for the current goal is (If_i_0 p x y H1).
∎
Definition. We define UPair to be λy z ⇒ {if Empty ∈ X then y else z|X ∈ 𝒫 (𝒫 Empty)} of type set → set → set.
In Proofgold the corresponding term root is 742438... and object id is dc3134...
Notation. {x,y} is notation for UPair x y.
Theorem. (UPairE)
∀x y z : set, x ∈ {y,z} → x = y ∨ x = z
In Proofgold the corresponding term root is 5009cc... and proposition id is e67fc5...
Proof:
Let x, y and z be given.
Assume H1: x ∈ {y,z}.
Apply (ReplE (𝒫 (𝒫 Empty)) (λX ⇒ if Empty ∈ X then y else z) x H1) to the current goal.
Let X be given.
Assume H2: X ∈ 𝒫 (𝒫 Empty) ∧ x = if Empty ∈ X then y else z.
We prove the intermediate claim L1: x = if Empty ∈ X then y else z.
An exact proof term for the current goal is (andER (X ∈ 𝒫 (𝒫 Empty)) (x = if Empty ∈ X then y else z) H2).
Apply (If_i_or (Empty ∈ X) y z) to the current goal.
Assume H3: (if Empty ∈ X then y else z) = y.
Apply orIL to the current goal.
We will prove x = y.
rewrite the current goal using H3 (from right to left).
An exact proof term for the current goal is L1.
Assume H3: (if Empty ∈ X then y else z) = z.
Apply orIR to the current goal.
We will prove x = z.
rewrite the current goal using H3 (from right to left).
An exact proof term for the current goal is L1.
∎
Theorem. (UPairI1)
∀y z : set, y ∈ {y,z}
In Proofgold the corresponding term root is b1eea2... and proposition id is bf3649...
Proof:
Let y and z be given.
We will prove y ∈ {y,z}.
rewrite the current goal using (If_i_1 (Empty ∈ 𝒫 Empty) y z (Empty_In_Power Empty)) (from right to left) at position 1.
We will prove (if Empty ∈ 𝒫 Empty then y else z) ∈ {y,z}.
We will prove (if Empty ∈ 𝒫 Empty then y else z) ∈ {if Empty ∈ X then y else z|X ∈ 𝒫 (𝒫 Empty)}.
Apply (ReplI (𝒫 (𝒫 Empty)) (λX : set ⇒ if (Empty ∈ X) then y else z) (𝒫 Empty)) to the current goal.
We will prove 𝒫 Empty ∈ 𝒫 (𝒫 Empty).
An exact proof term for the current goal is (Self_In_Power (𝒫 Empty)).
∎
Theorem. (UPairI2)
∀y z : set, z ∈ {y,z}
In Proofgold the corresponding term root is 27455b... and proposition id is a0a5b8...
Proof:
Let y and z be given.
We will prove z ∈ {y,z}.
rewrite the current goal using (If_i_0 (Empty ∈ Empty) y z (EmptyE Empty)) (from right to left) at position 1.
We will prove (if Empty ∈ Empty then y else z) ∈ {y,z}.
We will prove (if Empty ∈ Empty then y else z) ∈ {if Empty ∈ X then y else z|X ∈ 𝒫 (𝒫 Empty)}.
Apply (ReplI (𝒫 (𝒫 Empty)) (λX : set ⇒ if (Empty ∈ X) then y else z) Empty) to the current goal.
We will prove Empty ∈ 𝒫 (𝒫 Empty).
An exact proof term for the current goal is (Empty_In_Power (𝒫 Empty)).
∎
Definition. We define Sing to be λx ⇒ {x,x} of type set → set.
In Proofgold the corresponding term root is bd01a8... and object id is b7dfc1...
Notation. {x} is notation for Sing x.
Theorem. (SingI)
∀x : set, x ∈ {x}
In Proofgold the corresponding term root is 6591bd... and proposition id is 2f93be...
Proof:
An exact proof term for the current goal is (λx : set ⇒ UPairI1 x x).
∎
Theorem. (SingE)
∀x y : set, y ∈ {x} → y = x
In Proofgold the corresponding term root is 9341fe... and proposition id is 3b0dd8...
Proof:
An exact proof term for the current goal is (λx y H ⇒ UPairE y x x H (y = x) (λH ⇒ H) (λH ⇒ H)).
∎
Definition. We define binunion to be λX Y ⇒ ⋃ {X,Y} of type set → set → set.
In Proofgold the corresponding term root is 5e1ac4... and object id is e14989...
Notation. We use ∪ as an infix operator with priority 345 and which associates to the left corresponding to applying term binunion.
Theorem. (binunionI1)
∀X Y z : set, z ∈ X → z ∈ X ∪ Y
In Proofgold the corresponding term root is 753b6e... and proposition id is e43c8d...
Proof:
Let X, Y and z be given.
Assume H1: z ∈ X.
We will prove z ∈ X ∪ Y.
We will prove z ∈ ⋃ {X,Y}.
Apply (UnionI {X,Y} z X) to the current goal.
We will prove z ∈ X.
An exact proof term for the current goal is H1.
We will prove X ∈ {X,Y}.
An exact proof term for the current goal is (UPairI1 X Y).
∎
Theorem. (binunionI2)
∀X Y z : set, z ∈ Y → z ∈ X ∪ Y
In Proofgold the corresponding term root is b8821c... and proposition id is 959535...
Proof:
Let X, Y and z be given.
Assume H1: z ∈ Y.
We will prove z ∈ X ∪ Y.
We will prove z ∈ ⋃ {X,Y}.
Apply (UnionI {X,Y} z Y) to the current goal.
We will prove z ∈ Y.
An exact proof term for the current goal is H1.
We will prove Y ∈ {X,Y}.
An exact proof term for the current goal is (UPairI2 X Y).
∎
Theorem. (binunionE)
∀X Y z : set, z ∈ X ∪ Y → z ∈ X ∨ z ∈ Y
In Proofgold the corresponding term root is 5a3195... and proposition id is 1ceeb0...
Proof:
Let X, Y and z be given.
Assume H1: z ∈ X ∪ Y.
We will prove z ∈ X ∨ z ∈ Y.
Apply (UnionE_impred {X,Y} z H1) to the current goal.
Let Z be given.
Assume H2: z ∈ Z.
Assume H3: Z ∈ {X,Y}.
Apply (UPairE Z X Y H3) to the current goal.
Assume H4: Z = X.
Apply orIL to the current goal.
We will prove z ∈ X.
rewrite the current goal using H4 (from right to left).
We will prove z ∈ Z.
An exact proof term for the current goal is H2.
Assume H4: Z = Y.
Apply orIR to the current goal.
We will prove z ∈ Y.
rewrite the current goal using H4 (from right to left).
We will prove z ∈ Z.
An exact proof term for the current goal is H2.
∎
Theorem. (binunionE')
∀X Y z, ∀p : prop, (z ∈ X → p) → (z ∈ Y → p) → (z ∈ X ∪ Y → p)
In Proofgold the corresponding term root is e5b951... and proposition id is 570dd1...
Proof:
Let X, Y, z and p be given.
Assume H1 H2 Hz.
Apply binunionE X Y z Hz to the current goal.
Assume H3: z ∈ X.
An exact proof term for the current goal is H1 H3.
Assume H3: z ∈ Y.
An exact proof term for the current goal is H2 H3.
∎
Theorem. (binunion_asso)
∀X Y Z : set, X ∪ (Y ∪ Z) = (X ∪ Y) ∪ Z
In Proofgold the corresponding term root is e0f701... and proposition id is 61dfba...
Proof:
Let X, Y and Z be given.
Apply set_ext to the current goal.
Let w be given.
Assume H1: w ∈ X ∪ (Y ∪ Z).
We will prove w ∈ (X ∪ Y) ∪ Z.
Apply (binunionE X (Y ∪ Z) w H1) to the current goal.
Assume H2: w ∈ X.
Apply binunionI1 to the current goal.
Apply binunionI1 to the current goal.
An exact proof term for the current goal is H2.
Assume H2: w ∈ Y ∪ Z.
Apply (binunionE Y Z w H2) to the current goal.
Assume H3: w ∈ Y.
Apply binunionI1 to the current goal.
Apply binunionI2 to the current goal.
An exact proof term for the current goal is H3.
Assume H3: w ∈ Z.
Apply binunionI2 to the current goal.
An exact proof term for the current goal is H3.
Let w be given.
Assume H1: w ∈ (X ∪ Y) ∪ Z.
We will prove w ∈ X ∪ (Y ∪ Z).
Apply (binunionE (X ∪ Y) Z w H1) to the current goal.
Assume H2: w ∈ X ∪ Y.
Apply (binunionE X Y w H2) to the current goal.
Assume H3: w ∈ X.
Apply binunionI1 to the current goal.
An exact proof term for the current goal is H3.
Assume H3: w ∈ Y.
Apply binunionI2 to the current goal.
Apply binunionI1 to the current goal.
An exact proof term for the current goal is H3.
Assume H2: w ∈ Z.
Apply binunionI2 to the current goal.
Apply binunionI2 to the current goal.
An exact proof term for the current goal is H2.
∎
Theorem. (binunion_com_Subq)
∀X Y : set, X ∪ Y ⊆ Y ∪ X
In Proofgold the corresponding term root is ee8f5a... and proposition id is d84076...
Proof:
Let X, Y and w be given.
Assume H1: w ∈ X ∪ Y.
We will prove w ∈ Y ∪ X.
Apply (binunionE X Y w H1) to the current goal.
Assume H2: w ∈ X.
Apply binunionI2 to the current goal.
An exact proof term for the current goal is H2.
Assume H2: w ∈ Y.
Apply binunionI1 to the current goal.
An exact proof term for the current goal is H2.
∎
Theorem. (binunion_com)
∀X Y : set, X ∪ Y = Y ∪ X
In Proofgold the corresponding term root is e20f4f... and proposition id is bed33e...
Proof:
Let X and Y be given.
Apply set_ext to the current goal.
An exact proof term for the current goal is (binunion_com_Subq X Y).
An exact proof term for the current goal is (binunion_com_Subq Y X).
∎
Theorem. (binunion_idl)
∀X : set, Empty ∪ X = X
In Proofgold the corresponding term root is 8dc616... and proposition id is 52cd24...
Proof:
Let X be given.
Apply set_ext to the current goal.
Let x be given.
Assume H1: x ∈ Empty ∪ X.
Apply (binunionE Empty X x H1) to the current goal.
Assume H2: x ∈ Empty.
We will prove False.
An exact proof term for the current goal is (EmptyE x H2).
Assume H2: x ∈ X.
An exact proof term for the current goal is H2.
Let x be given.
Assume H2: x ∈ X.
We will prove x ∈ Empty ∪ X.
Apply binunionI2 to the current goal.
An exact proof term for the current goal is H2.
∎
Theorem. (binunion_idr)
∀X : set, X ∪ Empty = X
In Proofgold the corresponding term root is 67e82f... and proposition id is d5a005...
Proof:
Let X be given.
rewrite the current goal using (binunion_com X Empty) (from left to right).
An exact proof term for the current goal is (binunion_idl X).
∎
Theorem. (binunion_Subq_1)
∀X Y : set, X ⊆ X ∪ Y
In Proofgold the corresponding term root is b2c658... and proposition id is a9b3a8...
Proof:
An exact proof term for the current goal is binunionI1.
∎
Theorem. (binunion_Subq_2)
∀X Y : set, Y ⊆ X ∪ Y
In Proofgold the corresponding term root is 3cf229... and proposition id is 1507ce...
Proof:
An exact proof term for the current goal is binunionI2.
∎
Theorem. (binunion_Subq_min)
∀X Y Z : set, X ⊆ Z → Y ⊆ Z → X ∪ Y ⊆ Z
In Proofgold the corresponding term root is cc32a5... and proposition id is d64da4...
Proof:
Let X, Y and Z be given.
Assume H1: X ⊆ Z.
Assume H2: Y ⊆ Z.
Let w be given.
Assume H3: w ∈ X ∪ Y.
Apply (binunionE X Y w H3) to the current goal.
Assume H4: w ∈ X.
An exact proof term for the current goal is (H1 w H4).
Assume H4: w ∈ Y.
An exact proof term for the current goal is (H2 w H4).
∎
Theorem. (Subq_binunion_eq)
∀X Y, (X ⊆ Y) = (X ∪ Y = Y)
In Proofgold the corresponding term root is f8ee53... and proposition id is c2d393...
Proof:
Let X and Y be given.
Apply prop_ext_2 to the current goal.
Assume H1: X ⊆ Y.
We will prove X ∪ Y = Y.
Apply set_ext to the current goal.
We will prove X ∪ Y ⊆ Y.
Apply (binunion_Subq_min X Y Y) to the current goal.
We will prove X ⊆ Y.
An exact proof term for the current goal is H1.
We will prove Y ⊆ Y.
An exact proof term for the current goal is (Subq_ref Y).
We will prove Y ⊆ X ∪ Y.
An exact proof term for the current goal is (binunion_Subq_2 X Y).
Assume H1: X ∪ Y = Y.
We will prove X ⊆ Y.
rewrite the current goal using H1 (from right to left).
We will prove X ⊆ X ∪ Y.
An exact proof term for the current goal is (binunion_Subq_1 X Y).
∎
Definition. We define SetAdjoin to be λX y ⇒ X ∪ {y} of type set → set → set.
In Proofgold the corresponding term root is 1f3a09... and object id is 5bdda8...
Notation. We now use the set enumeration notation {...,...,...} in general. If 0 elements are given, then Empty is used to form the corresponding term. If 1 element is given, then Sing is used to form the corresponding term. If 2 elements are given, then UPair is used to form the corresponding term. If more than elements are given, then SetAdjoin is used to reduce to the case with one fewer elements.
Definition. We define famunion to be λX F ⇒ ⋃ {F x|x ∈ X} of type set → (set → set) → set.
In Proofgold the corresponding term root is b3e3bf... and object id is 241349...
Notation. We use ⋃ x [possibly with ascriptions] , B as a binder notation corresponding to a term constructed using famunion.
Theorem. (famunionI)
∀X : set, ∀F : (set → set), ∀x y : set, x ∈ X → y ∈ F x → y ∈ ⋃x ∈ XF x
In Proofgold the corresponding term root is 0c23d0... and proposition id is e4f490...
Proof:
An exact proof term for the current goal is (λX F x y H1 H2 ⇒ UnionI (Repl X F) y (F x) H2 (ReplI X F x H1)).
∎
Theorem. (famunionE)
∀X : set, ∀F : (set → set), ∀y : set, y ∈ (⋃x ∈ XF x) → ∃x ∈ X, y ∈ F x
In Proofgold the corresponding term root is 48683a... and proposition id is ede6e5...
Proof:
Let X, F and y be given.
Assume H1: y ∈ (⋃x ∈ XF x).
We will prove ∃x ∈ X, y ∈ F x.
Apply (UnionE_impred {F x|x ∈ X} y H1) to the current goal.
Let Y be given.
Assume H2: y ∈ Y.
Assume H3: Y ∈ {F x|x ∈ X}.
Apply (ReplE_impred X F Y H3) to the current goal.
Let x be given.
Assume H4: x ∈ X.
Assume H5: Y = F x.
We use x to witness the existential quantifier.
We will prove x ∈ X ∧ y ∈ F x.
Apply andI to the current goal.
An exact proof term for the current goal is H4.
We will prove y ∈ F x.
rewrite the current goal using H5 (from right to left).
An exact proof term for the current goal is H2.
∎
Theorem. (famunionE_impred)
∀X : set, ∀F : (set → set), ∀y : set, y ∈ (⋃x ∈ XF x) → ∀p : prop, (∀x, x ∈ X → y ∈ F x → p) → p
In Proofgold the corresponding term root is ab2266... and proposition id is 562023...
Proof:
Let X, F and y be given.
Assume Hy.
Let p be given.
Assume Hp.
Apply famunionE X F y Hy to the current goal.
Let x be given.
Assume H1.
Apply H1 to the current goal.
An exact proof term for the current goal is Hp x.
∎
Theorem. (famunion_Empty)
∀F : set → set, (⋃x ∈ EmptyF x) = Empty
In Proofgold the corresponding term root is fc1bdd... and proposition id is 22bd10...
Proof:
Let F be given.
Apply Empty_Subq_eq to the current goal.
Let y be given.
Assume Hy: y ∈ ⋃x ∈ EmptyF x.
Apply famunionE_impred Empty F y Hy to the current goal.
Let x be given.
Assume Hx: x ∈ Empty.
We will prove False.
An exact proof term for the current goal is EmptyE x Hx.
∎
Theorem. (famunion_Subq)
∀X, ∀f g : set → set, (∀x ∈ X, f x ⊆ g x) → famunion X f ⊆ famunion X g
In Proofgold the corresponding term root is 5f0b79... and proposition id is b1b158...
Proof:
Let X, f and g be given.
Assume Hfg.
Let y be given.
Assume Hy.
Apply famunionE_impred X f y Hy to the current goal.
Let x be given.
Assume Hx.
Assume H1: y ∈ f x.
Apply famunionI X g x y Hx to the current goal.
We will prove y ∈ g x.
An exact proof term for the current goal is Hfg x Hx y H1.
∎
Theorem. (famunion_ext)
∀X, ∀f g : set → set, (∀x ∈ X, f x = g x) → famunion X f = famunion X g
In Proofgold the corresponding term root is 8dc4c3... and proposition id is 5bc238...
Proof:
Let X, f and g be given.
Assume Hfg.
Apply set_ext to the current goal.
Apply famunion_Subq to the current goal.
Let x be given.
Assume Hx.
rewrite the current goal using Hfg x Hx (from left to right).
Apply Subq_ref to the current goal.
Apply famunion_Subq to the current goal.
Let x be given.
Assume Hx.
rewrite the current goal using Hfg x Hx (from left to right).
Apply Subq_ref to the current goal.
∎
Beginning of Section SepSec
Variable X : set
Variable P : set → prop
Let z : set ≝ Eps_i (λz ⇒ z ∈ X ∧ P z)
Let F : set → set ≝ λx ⇒ if P x then x else z
Definition. We define Sep to be if (∃z ∈ X, P z) then {F x|x ∈ X} else Empty of type set.
In Proofgold the corresponding term root is f336a4... and object id is 7008ec...
End of Section SepSec
Notation. {x ∈ A | B} is notation for Sep A (λ x . B).
Theorem. (SepI)
∀X : set, ∀P : (set → prop), ∀x : set, x ∈ X → P x → x ∈ {x ∈ X|P x}
In Proofgold the corresponding term root is ea56bb... and proposition id is 147a6a...
Proof:
Let X, P and x be given.
Set z to be the term Eps_i (λz ⇒ z ∈ X ∧ P z) of type set.
Set F to be the term λx ⇒ if P x then x else z of type set → set.
Assume H1: x ∈ X.
Assume H2: P x.
We prove the intermediate claim L1: ∃z ∈ X, P z.
We use x to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is H1.
An exact proof term for the current goal is H2.
We will prove x ∈ {x ∈ X|P x}.
We will prove x ∈ if (∃z ∈ X, P z) then {F x|x ∈ X} else Empty.
We prove the intermediate claim L2: (if (∃z ∈ X, P z) then {F x|x ∈ X} else Empty) = {F x|x ∈ X}.
An exact proof term for the current goal is (If_i_1 (∃z ∈ X, P z) {F x|x ∈ X} Empty L1).
rewrite the current goal using L2 (from left to right).
We will prove x ∈ {F x|x ∈ X}.
We prove the intermediate claim L3: F x = x.
We will prove (if P x then x else z) = x.
An exact proof term for the current goal is (If_i_1 (P x) x z H2).
rewrite the current goal using L3 (from right to left).
We will prove F x ∈ {F x|x ∈ X}.
An exact proof term for the current goal is (ReplI X F x H1).
∎
Theorem. (SepE)
∀X : set, ∀P : (set → prop), ∀x : set, x ∈ {x ∈ X|P x} → x ∈ X ∧ P x
In Proofgold the corresponding term root is 102830... and proposition id is 00899a...
Proof:
Let X, P and x be given.
Set z to be the term Eps_i (λz ⇒ z ∈ X ∧ P z) of type set.
Set F to be the term λx ⇒ if P x then x else z of type set → set.
Apply (xm (∃z ∈ X, P z)) to the current goal.
Assume H1: ∃z ∈ X, P z.
We will prove (x ∈ (if (∃z ∈ X, P z) then {F x|x ∈ X} else Empty) → x ∈ X ∧ P x).
We prove the intermediate claim L1: (if (∃z ∈ X, P z) then {F x|x ∈ X} else Empty) = {F x|x ∈ X}.
An exact proof term for the current goal is (If_i_1 (∃z ∈ X, P z) {F x|x ∈ X} Empty H1).
rewrite the current goal using L1 (from left to right).
We will prove x ∈ {F x|x ∈ X} → x ∈ X ∧ P x.
Assume H2: x ∈ {F x|x ∈ X}.
Apply (ReplE_impred X F x H2) to the current goal.
Let y be given.
Assume H3: y ∈ X.
Assume H4: x = F y.
We will prove x ∈ X ∧ P x.
Apply (xm (P y)) to the current goal.
Assume H5: P y.
We prove the intermediate claim L2: x = y.
rewrite the current goal using (If_i_1 (P y) y z H5) (from right to left).
An exact proof term for the current goal is H4.
rewrite the current goal using L2 (from left to right).
We will prove y ∈ X ∧ P y.
Apply andI to the current goal.
An exact proof term for the current goal is H3.
An exact proof term for the current goal is H5.
Assume H5: ¬ P y.
We prove the intermediate claim L2: x = z.
rewrite the current goal using (If_i_0 (P y) y z H5) (from right to left).
An exact proof term for the current goal is H4.
rewrite the current goal using L2 (from left to right).
We will prove z ∈ X ∧ P z.
An exact proof term for the current goal is (Eps_i_ex (λz ⇒ z ∈ X ∧ P z) H1).
Assume H1: ¬ ∃z ∈ X, P z.
We will prove (x ∈ (if (∃z ∈ X, P z) then {F x|x ∈ X} else Empty) → x ∈ X ∧ P x).
We prove the intermediate claim L1: (if (∃z ∈ X, P z) then {F x|x ∈ X} else Empty) = Empty.
An exact proof term for the current goal is (If_i_0 (∃z ∈ X, P z) {F x|x ∈ X} Empty H1).
rewrite the current goal using L1 (from left to right).
We will prove x ∈ Empty → x ∈ X ∧ P x.
Assume H2: x ∈ Empty.
We will prove False.
An exact proof term for the current goal is (EmptyE x H2).
∎
Theorem. (SepE1)
∀X : set, ∀P : (set → prop), ∀x : set, x ∈ {x ∈ X|P x} → x ∈ X
In Proofgold the corresponding term root is d83445... and proposition id is 6399b0...
Proof:
An exact proof term for the current goal is (λX P x H ⇒ SepE X P x H (x ∈ X) (λH _ ⇒ H)).
∎
Theorem. (SepE2)
∀X : set, ∀P : (set → prop), ∀x : set, x ∈ {x ∈ X|P x} → P x
In Proofgold the corresponding term root is aa61ca... and proposition id is 104227...
Proof:
An exact proof term for the current goal is (λX P x H ⇒ SepE X P x H (P x) (λ_ H ⇒ H)).
∎
Theorem. (Sep_Empty)
∀P : set → prop, {x ∈ Empty|P x} = Empty
In Proofgold the corresponding term root is f36949... and proposition id is e3473a...
Proof:
Let P be given.
Apply Empty_eq to the current goal.
Let x be given.
Assume Hx.
An exact proof term for the current goal is EmptyE x (SepE1 Empty P x Hx).
∎
Theorem. (Sep_Subq)
∀X : set, ∀P : set → prop, {x ∈ X|P x} ⊆ X
In Proofgold the corresponding term root is 97d080... and proposition id is c17d45...
Proof:
An exact proof term for the current goal is SepE1.
∎
Theorem. (Sep_In_Power)
∀X : set, ∀P : set → prop, {x ∈ X|P x} ∈ 𝒫 X
In Proofgold the corresponding term root is 612c94... and proposition id is fba53e...
Proof:
An exact proof term for the current goal is (λX P ⇒ PowerI X (Sep X P) (Sep_Subq X P)).
∎
Definition. We define ReplSep to be λX P F ⇒ {F x|x ∈ {z ∈ X|P z}} of type set → (set → prop) → (set → set) → set.
In Proofgold the corresponding term root is ec807b... and object id is 9bdb56...
Notation. {B| x ∈ A, C} is notation for ReplSep A (λ x . C) (λ x . B).
Theorem. (ReplSepI)
∀X : set, ∀P : set → prop, ∀F : set → set, ∀x : set, x ∈ X → P x → F x ∈ {F x|x ∈ X, P x}
In Proofgold the corresponding term root is 35b8ec... and proposition id is aa7b2b...
Proof:
An exact proof term for the current goal is (λX P F x u v ⇒ ReplI (Sep X P) F x (SepI X P x u v)).
∎
Theorem. (ReplSepE)
∀X : set, ∀P : set → prop, ∀F : set → set, ∀y : set, y ∈ {F x|x ∈ X, P x} → ∃x : set, x ∈ X ∧ P x ∧ y = F x
In Proofgold the corresponding term root is 415ff1... and proposition id is b1bc76...
Proof:
Let X, P, F and y be given.
Assume H1: y ∈ {F x|x ∈ {z ∈ X|P z}}.
Apply (ReplE {z ∈ X|P z} F y H1) to the current goal.
Let x be given.
Assume H2: x ∈ {z ∈ X|P z} ∧ y = F x.
Apply H2 to the current goal.
Assume H3: x ∈ {z ∈ X|P z}.
Assume H4: y = F x.
Apply (SepE X P x H3) to the current goal.
Assume H5: x ∈ X.
Assume H6: P x.
We use x to witness the existential quantifier.
Apply and3I to the current goal.
An exact proof term for the current goal is H5.
An exact proof term for the current goal is H6.
An exact proof term for the current goal is H4.
∎
Theorem. (ReplSepE_impred)
∀X : set, ∀P : set → prop, ∀F : set → set, ∀y : set, y ∈ {F x|x ∈ X, P x} → ∀p : prop, (∀x ∈ X, P x → y = F x → p) → p
In Proofgold the corresponding term root is 5d26e1... and proposition id is 063cea...
Proof:
Let X, P, F and y be given.
Assume H1: y ∈ {F x|x ∈ X, P x}.
Let p be given.
Assume H2: ∀x ∈ X, P x → y = F x → p.
We will prove p.
Apply ReplSepE X P F y H1 to the current goal.
Let x be given.
Assume H3.
Apply H3 to the current goal.
Assume H3.
Apply H3 to the current goal.
An exact proof term for the current goal is H2 x.
∎
Definition. We define binintersect to be λX Y ⇒ {x ∈ X|x ∈ Y} of type set → set → set.
In Proofgold the corresponding term root is b2abd2... and object id is 2c68d8...
Notation. We use ∩ as an infix operator with priority 340 and which associates to the left corresponding to applying term binintersect.
Theorem. (binintersectI)
∀X Y z, z ∈ X → z ∈ Y → z ∈ X ∩ Y
In Proofgold the corresponding term root is fd656d... and proposition id is 9081a0...
Proof:
An exact proof term for the current goal is (λX Y z H1 H2 ⇒ SepI X (λx : set ⇒ x ∈ Y) z H1 H2).
∎
Theorem. (binintersectE)
∀X Y z, z ∈ X ∩ Y → z ∈ X ∧ z ∈ Y
In Proofgold the corresponding term root is 463b5a... and proposition id is ca0636...
Proof:
An exact proof term for the current goal is (λX Y z H1 ⇒ SepE X (λx : set ⇒ x ∈ Y) z H1).
∎
Theorem. (binintersectE1)
∀X Y z, z ∈ X ∩ Y → z ∈ X
In Proofgold the corresponding term root is b12123... and proposition id is 51005e...
Proof:
An exact proof term for the current goal is (λX Y z H1 ⇒ SepE1 X (λx : set ⇒ x ∈ Y) z H1).
∎
Theorem. (binintersectE2)
∀X Y z, z ∈ X ∩ Y → z ∈ Y
In Proofgold the corresponding term root is 39656f... and proposition id is 599de4...
Proof:
An exact proof term for the current goal is (λX Y z H1 ⇒ SepE2 X (λx : set ⇒ x ∈ Y) z H1).
∎
Theorem. (binintersect_Subq_1)
∀X Y : set, X ∩ Y ⊆ X
In Proofgold the corresponding term root is f296ad... and proposition id is 303833...
Proof:
An exact proof term for the current goal is binintersectE1.
∎
Theorem. (binintersect_Subq_2)
∀X Y : set, X ∩ Y ⊆ Y
In Proofgold the corresponding term root is fcd5f5... and proposition id is 8b9fd2...
Proof:
An exact proof term for the current goal is binintersectE2.
∎
Theorem. (binintersect_Subq_eq_1)
∀X Y, X ⊆ Y → X ∩ Y = X
In Proofgold the corresponding term root is 277eec... and proposition id is ec1cc3...
Proof:
Let X and Y be given.
Assume H1: X ⊆ Y.
Apply set_ext to the current goal.
Apply binintersect_Subq_1 to the current goal.
Let x be given.
Assume H2: x ∈ X.
Apply binintersectI to the current goal.
An exact proof term for the current goal is H2.
Apply H1 to the current goal.
An exact proof term for the current goal is H2.
∎
Theorem. (binintersect_Subq_max)
∀X Y Z : set, Z ⊆ X → Z ⊆ Y → Z ⊆ X ∩ Y
In Proofgold the corresponding term root is 95624d... and proposition id is 64636b...
Proof:
Let X, Y and Z be given.
Assume H1: Z ⊆ X.
Assume H2: Z ⊆ Y.
Let w be given.
Assume H3: w ∈ Z.
Apply (binintersectI X Y w) to the current goal.
We will prove w ∈ X.
An exact proof term for the current goal is (H1 w H3).
We will prove w ∈ Y.
An exact proof term for the current goal is (H2 w H3).
∎
Theorem. (binintersect_com_Subq)
∀X Y : set, X ∩ Y ⊆ Y ∩ X
In Proofgold the corresponding term root is 410255... and proposition id is 5977b8...
Proof:
Let X and Y be given.
Apply (binintersect_Subq_max Y X (X ∩ Y)) to the current goal.
We will prove X ∩ Y ⊆ Y.
Apply binintersect_Subq_2 to the current goal.
We will prove X ∩ Y ⊆ X.
Apply binintersect_Subq_1 to the current goal.
∎
Theorem. (binintersect_com)
∀X Y : set, X ∩ Y = Y ∩ X
In Proofgold the corresponding term root is 1a2c6f... and proposition id is ea8a1c...
Proof:
Let X and Y be given.
Apply set_ext to the current goal.
An exact proof term for the current goal is (binintersect_com_Subq X Y).
An exact proof term for the current goal is (binintersect_com_Subq Y X).
∎
Definition. We define setminus to be λX Y ⇒ Sep X (λx ⇒ x ∉ Y) of type set → set → set.
In Proofgold the corresponding term root is c68e5a... and object id is 913da1...
Notation. We use ∖ as an infix operator with priority 350 and no associativity corresponding to applying term setminus.
Theorem. (setminusI)
∀X Y z, (z ∈ X) → (z ∉ Y) → z ∈ X ∖ Y
In Proofgold the corresponding term root is 0feeb8... and proposition id is 80db6a...
Proof:
An exact proof term for the current goal is (λX Y z H1 H2 ⇒ SepI X (λx : set ⇒ x ∉ Y) z H1 H2).
∎
Theorem. (setminusE)
∀X Y z, (z ∈ X ∖ Y) → z ∈ X ∧ z ∉ Y
In Proofgold the corresponding term root is 6774f4... and proposition id is 71763a...
Proof:
An exact proof term for the current goal is (λX Y z H ⇒ SepE X (λx : set ⇒ x ∉ Y) z H).
∎
Theorem. (setminusE1)
∀X Y z, (z ∈ X ∖ Y) → z ∈ X
In Proofgold the corresponding term root is 064bd5... and proposition id is d74baf...
Proof:
An exact proof term for the current goal is (λX Y z H ⇒ SepE1 X (λx : set ⇒ x ∉ Y) z H).
∎
Theorem. (setminusE2)
∀X Y z, (z ∈ X ∖ Y) → z ∉ Y
In Proofgold the corresponding term root is 3e6969... and proposition id is 36b273...
Proof:
An exact proof term for the current goal is (λX Y z H ⇒ SepE2 X (λx : set ⇒ x ∉ Y) z H).
∎
Theorem. (setminus_Subq)
∀X Y : set, X ∖ Y ⊆ X
In Proofgold the corresponding term root is 5a497a... and proposition id is 66f982...
Proof:
An exact proof term for the current goal is setminusE1.
∎
Theorem. (setminus_In_Power)
∀A U, A ∖ U ∈ 𝒫 A
In Proofgold the corresponding term root is 062275... and proposition id is f503e1...
Proof:
Let A and U be given.
Apply PowerI to the current goal.
Apply setminus_Subq to the current goal.
∎
Theorem. (binunion_remove1_eq)
∀X, ∀x ∈ X, X = (X ∖ {x}) ∪ {x}
In Proofgold the corresponding term root is 20fce6... and proposition id is eb0c40...
Proof:
Let X and x be given.
Assume Hx: x ∈ X.
Apply set_ext to the current goal.
Let y be given.
Assume Hy: y ∈ X.
We will prove y ∈ (X ∖ {x}) ∪ {x}.
Apply xm (y ∈ {x}) to the current goal.
Assume H1: y ∈ {x}.
Apply binunionI2 to the current goal.
An exact proof term for the current goal is H1.
Assume H1: y ∉ {x}.
Apply binunionI1 to the current goal.
Apply setminusI to the current goal.
An exact proof term for the current goal is Hy.
An exact proof term for the current goal is H1.
Let y be given.
Assume Hy: y ∈ (X ∖ {x}) ∪ {x}.
Apply binunionE (X ∖ {x}) {x} y Hy to the current goal.
Assume H1: y ∈ X ∖ {x}.
We will prove y ∈ X.
An exact proof term for the current goal is setminusE1 X {x} y H1.
Assume H1: y ∈ {x}.
We will prove y ∈ X.
rewrite the current goal using SingE x y H1 (from left to right).
We will prove x ∈ X.
An exact proof term for the current goal is Hx.
∎
Theorem. (In_irref)
∀x, x ∉ x
In Proofgold the corresponding term root is 12629a... and proposition id is 7bfd46...
Proof:
Apply In_ind to the current goal.
We will prove (∀X : set, (∀x : set, x ∈ X → x ∉ x) → X ∉ X).
Let X be given.
Assume IH: ∀x : set, x ∈ X → x ∉ x.
Assume H: X ∈ X.
An exact proof term for the current goal is IH X H H.
∎
Theorem. (In_no2cycle)
∀x y, x ∈ y → y ∈ x → False
In Proofgold the corresponding term root is 6d0cb3... and proposition id is 344f30...
Proof:
Apply In_ind to the current goal.
Let x be given.
Assume IH: ∀z, z ∈ x → ∀y, z ∈ y → y ∈ z → False.
Let y be given.
Assume H1: x ∈ y.
Assume H2: y ∈ x.
An exact proof term for the current goal is IH y H2 x H2 H1.
∎
Definition. We define ordsucc to be λx : set ⇒ x ∪ {x} of type set → set.
In Proofgold the corresponding term root is 65d883... and object id is 1452f7...
Theorem. (ordsuccI1)
∀x : set, x ⊆ ordsucc x
In Proofgold the corresponding term root is 5a2f15... and proposition id is 367d80...
Proof:
Let x be given.
An exact proof term for the current goal is (λ(y : set)(H1 : y ∈ x) ⇒ binunionI1 x {x} y H1).
∎
Theorem. (ordsuccI2)
∀x : set, x ∈ ordsucc x
In Proofgold the corresponding term root is 2f6efd... and proposition id is 2be6be...
Proof:
An exact proof term for the current goal is (λx : set ⇒ binunionI2 x {x} x (SingI x)).
∎
Theorem. (ordsuccE)
∀x y : set, y ∈ ordsucc x → y ∈ x ∨ y = x
In Proofgold the corresponding term root is 01c4ee... and proposition id is 3849f5...
Proof:
Let x and y be given.
Assume H1: y ∈ x ∪ {x}.
Apply (binunionE x {x} y H1) to the current goal.
Assume H2: y ∈ x.
Apply orIL to the current goal.
An exact proof term for the current goal is H2.
Assume H2: y ∈ {x}.
Apply orIR to the current goal.
An exact proof term for the current goal is (SingE x y H2).
∎
Notation. Natural numbers 0,1,2,... are notation for the terms formed using Empty as 0 and forming successors with ordsucc.
Theorem. (neq_0_ordsucc)
∀a : set, 0 ≠ ordsucc a
In Proofgold the corresponding term root is 1b20cb... and proposition id is 04c410...
Proof:
Let a be given.
We will prove ¬ (0 = ordsucc a).
Assume H1: 0 = ordsucc a.
We prove the intermediate claim L1: a ∈ ordsucc a → False.
rewrite the current goal using H1 (from right to left).
An exact proof term for the current goal is (EmptyE a).
An exact proof term for the current goal is (L1 (ordsuccI2 a)).
∎
Theorem. (neq_ordsucc_0)
∀a : set, ordsucc a ≠ 0
In Proofgold the corresponding term root is 9b391e... and proposition id is 346ee0...
Proof:
Let a be given.
An exact proof term for the current goal is neq_i_sym 0 (ordsucc a) (neq_0_ordsucc a).
∎
Theorem. (ordsucc_inj)
∀a b : set, ordsucc a = ordsucc b → a = b
In Proofgold the corresponding term root is 40cad6... and proposition id is 4869a1...
Proof:
Let a and b be given.
Assume H1: ordsucc a = ordsucc b.
We prove the intermediate claim L1: a ∈ ordsucc b.
rewrite the current goal using H1 (from right to left).
An exact proof term for the current goal is (ordsuccI2 a).
Apply (ordsuccE b a L1) to the current goal.
Assume H2: a ∈ b.
We prove the intermediate claim L2: b ∈ ordsucc a.
rewrite the current goal using H1 (from left to right).
An exact proof term for the current goal is (ordsuccI2 b).
Apply (ordsuccE a b L2) to the current goal.
Assume H3: b ∈ a.
We will prove False.
An exact proof term for the current goal is (In_no2cycle a b H2 H3).
Assume H3: b = a.
Use symmetry.
An exact proof term for the current goal is H3.
Assume H2: a = b.
An exact proof term for the current goal is H2.
∎
Theorem. (In_0_1)
0 ∈ 1
In Proofgold the corresponding term root is e984b0... and proposition id is 3be80a...
Proof:
An exact proof term for the current goal is (ordsuccI2 0).
∎
Theorem. (In_0_2)
0 ∈ 2
In Proofgold the corresponding term root is 56dc2c... and proposition id is 5a202e...
Proof:
An exact proof term for the current goal is (ordsuccI1 1 0 In_0_1).
∎
Theorem. (In_1_2)
1 ∈ 2
In Proofgold the corresponding term root is 8704b3... and proposition id is 431e9e...
Proof:
An exact proof term for the current goal is (ordsuccI2 1).
∎
Definition. We define nat_p to be λn : set ⇒ ∀p : set → prop, p 0 → (∀x : set, p x → p (ordsucc x)) → p n of type set → prop.
In Proofgold the corresponding term root is 458be3... and object id is cfd8eb...
Theorem. (nat_0)
nat_p 0
In Proofgold the corresponding term root is 1617bc... and proposition id is 457a72...
Proof:
An exact proof term for the current goal is (λp H _ ⇒ H).
∎
Theorem. (nat_ordsucc)
∀n : set, nat_p n → nat_p (ordsucc n)
In Proofgold the corresponding term root is 87f001... and proposition id is 57e940...
Proof:
An exact proof term for the current goal is (λn H1 p H2 H3 ⇒ H3 n (H1 p H2 H3)).
∎
Theorem. (nat_1)
nat_p 1
In Proofgold the corresponding term root is 2f83fa... and proposition id is 672d0d...
Proof:
An exact proof term for the current goal is (nat_ordsucc 0 nat_0).
∎
Theorem. (nat_2)
nat_p 2
In Proofgold the corresponding term root is a8be15... and proposition id is 9112e9...
Proof:
An exact proof term for the current goal is (nat_ordsucc 1 nat_1).
∎
Theorem. (nat_0_in_ordsucc)
∀n, nat_p n → 0 ∈ ordsucc n
In Proofgold the corresponding term root is 9b50b9... and proposition id is 251f3d...
Proof:
Let n be given.
Assume H1.
Apply H1 (λn ⇒ 0 ∈ ordsucc n) to the current goal.
We will prove 0 ∈ ordsucc 0.
An exact proof term for the current goal is In_0_1.
Let n be given.
Assume IH: 0 ∈ ordsucc n.
We will prove 0 ∈ ordsucc (ordsucc n).
An exact proof term for the current goal is (ordsuccI1 (ordsucc n) 0 IH).
∎
Theorem. (nat_ordsucc_in_ordsucc)
∀n, nat_p n → ∀m ∈ n, ordsucc m ∈ ordsucc n
In Proofgold the corresponding term root is 819399... and proposition id is 4c2c95...
Proof:
Let n be given.
Assume H1.
Apply (H1 (λn ⇒ ∀m ∈ n, ordsucc m ∈ ordsucc n)) to the current goal.
We will prove ∀m ∈ 0, ordsucc m ∈ ordsucc 0.
Let m be given.
Assume Hm: m ∈ 0.
We will prove False.
An exact proof term for the current goal is (EmptyE m Hm).
Let n be given.
Assume IH: ∀m ∈ n, ordsucc m ∈ ordsucc n.
We will prove ∀m ∈ ordsucc n, ordsucc m ∈ ordsucc (ordsucc n).
Let m be given.
Assume H2: m ∈ ordsucc n.
We will prove ordsucc m ∈ ordsucc (ordsucc n).
Apply (ordsuccE n m H2) to the current goal.
Assume H3: m ∈ n.
We prove the intermediate claim L1: ordsucc m ∈ ordsucc n.
An exact proof term for the current goal is (IH m H3).
An exact proof term for the current goal is (ordsuccI1 (ordsucc n) (ordsucc m) L1).
Assume H3: m = n.
rewrite the current goal using H3 (from left to right).
We will prove ordsucc n ∈ ordsucc (ordsucc n).
An exact proof term for the current goal is (ordsuccI2 (ordsucc n)).
∎
Theorem. (nat_ind)
∀p : set → prop, p 0 → (∀n, nat_p n → p n → p (ordsucc n)) → ∀n, nat_p n → p n
In Proofgold the corresponding term root is 5d06f5... and proposition id is ecdce8...
Proof:
Let p be given.
Assume H1: p 0.
Assume H2: ∀n, nat_p n → p n → p (ordsucc n).
We prove the intermediate claim L1: nat_p 0 ∧ p 0.
An exact proof term for the current goal is (andI (nat_p 0) (p 0) nat_0 H1).
We prove the intermediate claim L2: ∀n, nat_p n ∧ p n → nat_p (ordsucc n) ∧ p (ordsucc n).
Let n be given.
Assume H3: nat_p n ∧ p n.
Apply H3 to the current goal.
Assume H4: nat_p n.
Assume H5: p n.
Apply andI to the current goal.
We will prove nat_p (ordsucc n).
An exact proof term for the current goal is (nat_ordsucc n H4).
We will prove p (ordsucc n).
An exact proof term for the current goal is (H2 n H4 H5).
Let n be given.
Assume H3.
We prove the intermediate claim L3: nat_p n ∧ p n.
An exact proof term for the current goal is (H3 (λn ⇒ nat_p n ∧ p n) L1 L2).
An exact proof term for the current goal is (andER (nat_p n) (p n) L3).
∎
Theorem. (nat_complete_ind)
∀p : set → prop, (∀n, nat_p n → (∀m ∈ n, p m) → p n) → ∀n, nat_p n → p n
In Proofgold the corresponding term root is d32e5b... and proposition id is 1e1608...
Proof:
Let p be given.
Assume H1: ∀n, nat_p n → (∀m ∈ n, p m) → p n.
We prove the intermediate claim L1: ∀n : set, nat_p n → ∀m ∈ n, p m.
Apply nat_ind to the current goal.
We will prove ∀m ∈ 0, p m.
Let m be given.
Assume Hm: m ∈ 0.
We will prove False.
An exact proof term for the current goal is (EmptyE m Hm).
Let n be given.
Assume Hn: nat_p n.
Assume IHn: ∀m ∈ n, p m.
We will prove ∀m ∈ ordsucc n, p m.
Let m be given.
Assume Hm: m ∈ ordsucc n.
We will prove p m.
Apply (ordsuccE n m Hm) to the current goal.
Assume H2: m ∈ n.
An exact proof term for the current goal is (IHn m H2).
Assume H2: m = n.
We will prove p m.
rewrite the current goal using H2 (from left to right).
We will prove p n.
An exact proof term for the current goal is (H1 n Hn IHn).
We will prove ∀n, nat_p n → p n.
An exact proof term for the current goal is (λn Hn ⇒ H1 n Hn (L1 n Hn)).
∎
Theorem. (nat_inv_impred)
∀p : set → prop, p 0 → (∀n, nat_p n → p (ordsucc n)) → ∀n, nat_p n → p n
In Proofgold the corresponding term root is eddc5f... and proposition id is 175e22...
Proof:
Let p be given.
Assume H1 H2.
An exact proof term for the current goal is nat_ind p H1 (λn H _ ⇒ H2 n H).
∎
Theorem. (nat_inv)
∀n, nat_p n → n = 0 ∨ ∃x, nat_p x ∧ n = ordsucc x
In Proofgold the corresponding term root is 5d4296... and proposition id is 4f4167...
Proof:
Apply nat_inv_impred to the current goal.
Apply orIL to the current goal.
Use reflexivity.
Let n be given.
Assume Hn.
Apply orIR to the current goal.
We use n to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is Hn.
Use reflexivity.
∎
Theorem. (nat_p_trans)
∀n, nat_p n → ∀m ∈ n, nat_p m
In Proofgold the corresponding term root is e88449... and proposition id is 03b874...
Proof:
Apply nat_ind to the current goal.
We will prove ∀m ∈ 0, nat_p m.
Let m be given.
Assume Hm: m ∈ 0.
We will prove False.
An exact proof term for the current goal is (EmptyE m Hm).
Let n be given.
Assume Hn: nat_p n.
Assume IHn: ∀m ∈ n, nat_p m.
We will prove ∀m ∈ ordsucc n, nat_p m.
Let m be given.
Assume Hm: m ∈ ordsucc n.
Apply (ordsuccE n m Hm) to the current goal.
Assume H1: m ∈ n.
An exact proof term for the current goal is (IHn m H1).
Assume H1: m = n.
rewrite the current goal using H1 (from left to right).
An exact proof term for the current goal is Hn.
∎
Theorem. (nat_trans)
∀n, nat_p n → ∀m ∈ n, m ⊆ n
In Proofgold the corresponding term root is 6bc6e1... and proposition id is cad6fe...
Proof:
Apply nat_ind to the current goal.
We will prove ∀m ∈ 0, m ⊆ 0.
Let m be given.
Assume Hm: m ∈ 0.
We will prove False.
An exact proof term for the current goal is (EmptyE m Hm).
Let n be given.
Assume Hn: nat_p n.
Assume IHn: ∀m ∈ n, m ⊆ n.
We will prove ∀m ∈ ordsucc n, m ⊆ ordsucc n.
Let m be given.
Assume Hm: m ∈ ordsucc n.
Apply (ordsuccE n m Hm) to the current goal.
Assume H1: m ∈ n.
We will prove m ⊆ ordsucc n.
Apply (Subq_tra m n (ordsucc n)) to the current goal.
An exact proof term for the current goal is (IHn m H1).
An exact proof term for the current goal is (ordsuccI1 n).
Assume H1: m = n.
We will prove m ⊆ ordsucc n.
rewrite the current goal using H1 (from left to right).
We will prove n ⊆ ordsucc n.
An exact proof term for the current goal is (ordsuccI1 n).
∎
Theorem. (nat_ordsucc_trans)
∀n, nat_p n → ∀m ∈ ordsucc n, m ⊆ n
In Proofgold the corresponding term root is e42e58... and proposition id is bc868d...
Proof:
Let n be given.
Assume Hn: nat_p n.
Let m be given.
Assume Hm: m ∈ ordsucc n.
Let k be given.
Assume Hk: k ∈ m.
We will prove k ∈ n.
Apply (ordsuccE n m Hm) to the current goal.
Assume H1: m ∈ n.
An exact proof term for the current goal is nat_trans n Hn m H1 k Hk.
Assume H1: m = n.
rewrite the current goal using H1 (from right to left).
An exact proof term for the current goal is Hk.
∎
Definition. We define surj to be λX Y f ⇒ (∀u ∈ X, f u ∈ Y) ∧ (∀w ∈ Y, ∃u ∈ X, f u = w) of type set → set → (set → set) → prop.
In Proofgold the corresponding term root is 4b68e4... and object id is f022f9...
Theorem. (form100_63_surjCantor)
∀A : set, ∀f : set → set, ¬ surj A (𝒫 A) f
In Proofgold the corresponding term root is eb4dfa... and proposition id is feecd4...
Proof:
Let A and f be given.
Assume H.
Apply H to the current goal.
Assume H1: ∀u ∈ A, f u ∈ 𝒫 A.
Assume H2: ∀w ∈ 𝒫 A, ∃u ∈ A, f u = w.
Set D to be the term {x ∈ A|x ∉ f x}.
We prove the intermediate claim L1: D ∈ 𝒫 A.
An exact proof term for the current goal is Sep_In_Power A (λx ⇒ x ∉ f x).
Apply H2 D L1 to the current goal.
Let d be given.
Assume H.
Apply H to the current goal.
Assume Hd: d ∈ A.
Assume HfdD: f d = D.
We prove the intermediate claim L2: d ∉ D.
Assume H3: d ∈ D.
Apply SepE2 A (λx ⇒ x ∉ f x) d H3 to the current goal.
We will prove d ∈ f d.
rewrite the current goal using HfdD (from left to right).
We will prove d ∈ D.
An exact proof term for the current goal is H3.
Apply L2 to the current goal.
We will prove d ∈ D.
Apply SepI to the current goal.
We will prove d ∈ A.
An exact proof term for the current goal is Hd.
We will prove d ∉ f d.
rewrite the current goal using HfdD (from left to right).
An exact proof term for the current goal is L2.
∎
Definition. We define inj to be λX Y f ⇒ (∀u ∈ X, f u ∈ Y) ∧ (∀u v ∈ X, f u = f v → u = v) of type set → set → (set → set) → prop.
In Proofgold the corresponding term root is 264b04... and object id is 442958...
Theorem. (form100_63_injCantor)
∀A : set, ∀f : set → set, ¬ inj (𝒫 A) A f
In Proofgold the corresponding term root is 4f3dc8... and proposition id is 2f2854...
Proof:
Let A and f be given.
Assume H.
Apply H to the current goal.
Assume H1: ∀X ∈ 𝒫 A, f X ∈ A.
Assume H2: ∀X Y ∈ 𝒫 A, f X = f Y → X = Y.
Set D to be the term {f X|X ∈ 𝒫 A, f X ∉ X}.
We prove the intermediate claim L1: D ∈ 𝒫 A.
Apply PowerI to the current goal.
Let n be given.
Assume H3: n ∈ D.
Apply ReplSepE_impred (𝒫 A) (λX ⇒ f X ∉ X) f n H3 to the current goal.
Let X be given.
Assume HX: X ∈ 𝒫 A.
Assume H4: f X ∉ X.
Assume H5: n = f X.
We will prove n ∈ A.
rewrite the current goal using H5 (from left to right).
Apply H1 to the current goal.
An exact proof term for the current goal is HX.
We prove the intermediate claim L2: f D ∉ D.
Assume H3: f D ∈ D.
Apply ReplSepE_impred (𝒫 A) (λX ⇒ f X ∉ X) f (f D) H3 to the current goal.
Let X be given.
Assume HX: X ∈ 𝒫 A.
Assume H4: f X ∉ X.
Assume H5: f D = f X.
We prove the intermediate claim L2a: D = X.
An exact proof term for the current goal is H2 D L1 X HX H5.
Apply H4 to the current goal.
rewrite the current goal using L2a (from right to left).
An exact proof term for the current goal is H3.
Apply L2 to the current goal.
We will prove f D ∈ D.
Apply ReplSepI to the current goal.
We will prove D ∈ 𝒫 A.
An exact proof term for the current goal is L1.
We will prove f D ∉ D.
An exact proof term for the current goal is L2.
∎
Theorem. (injI)
∀X Y, ∀f : set → set, (∀x ∈ X, f x ∈ Y) → (∀x z ∈ X, f x = f z → x = z) → inj X Y f
In Proofgold the corresponding term root is 8739d9... and proposition id is 71c936...
Proof:
Let X, Y and f be given.
Assume H1 H2.
We will prove (∀x ∈ X, f x ∈ Y) ∧ (∀x z ∈ X, f x = f z → x = z).
Apply andI to the current goal.
An exact proof term for the current goal is H1.
An exact proof term for the current goal is H2.
∎
Theorem. (inj_comp)
∀X Y Z : set, ∀f g : set → set, inj X Y f → inj Y Z g → inj X Z (λx ⇒ g (f x))
In Proofgold the corresponding term root is 649bd9... and proposition id is 05a66c...
Proof:
Let X, Y, Z, f and g be given.
Assume Hf.
Assume Hg.
Apply Hf to the current goal.
Assume Hf1 Hf2.
Apply Hg to the current goal.
Assume Hg1 Hg2.
Apply injI to the current goal.
Let u be given.
Assume Hu: u ∈ X.
An exact proof term for the current goal is (Hg1 (f u) (Hf1 u Hu)).
Let u be given.
Assume Hu: u ∈ X.
Let v be given.
Assume Hv: v ∈ X.
Assume H1: g (f u) = g (f v).
We will prove u = v.
Apply Hf2 u Hu v Hv to the current goal.
We will prove f u = f v.
Apply Hg2 (f u) (Hf1 u Hu) (f v) (Hf1 v Hv) to the current goal.
We will prove g (f u) = g (f v).
An exact proof term for the current goal is H1.
∎
Definition. We define bij to be λX Y f ⇒ (∀u ∈ X, f u ∈ Y) ∧ (∀u v ∈ X, f u = f v → u = v) ∧ (∀w ∈ Y, ∃u ∈ X, f u = w) of type set → set → (set → set) → prop.
In Proofgold the corresponding term root is 76bef6... and object id is f7d93b...
Theorem. (bijI)
∀X Y, ∀f : set → set, (∀u ∈ X, f u ∈ Y) → (∀u v ∈ X, f u = f v → u = v) → (∀w ∈ Y, ∃u ∈ X, f u = w) → bij X Y f
In Proofgold the corresponding term root is a9064f... and proposition id is 3229f9...
Proof:
Let X, Y and f be given.
Assume Hf1 Hf2 Hf3.
We will prove (∀u ∈ X, f u ∈ Y) ∧ (∀u v ∈ X, f u = f v → u = v) ∧ (∀w ∈ Y, ∃u ∈ X, f u = w).
Apply and3I to the current goal.
An exact proof term for the current goal is Hf1.
An exact proof term for the current goal is Hf2.
An exact proof term for the current goal is Hf3.
∎
Theorem. (bijE)
∀X Y, ∀f : set → set, bij X Y f → ∀p : prop, ((∀u ∈ X, f u ∈ Y) → (∀u v ∈ X, f u = f v → u = v) → (∀w ∈ Y, ∃u ∈ X, f u = w) → p) → p
In Proofgold the corresponding term root is 2cf278... and proposition id is cf0759...
Proof:
Let X, Y and f be given.
Assume Hf.
Let p be given.
Assume Hp.
Apply Hf to the current goal.
Assume Hf.
Apply Hf to the current goal.
Assume Hf1 Hf2 Hf3.
An exact proof term for the current goal is Hp Hf1 Hf2 Hf3.
∎
Theorem. (bij_inj)
∀X Y, ∀f : set → set, bij X Y f → inj X Y f
In Proofgold the corresponding term root is e3635e... and proposition id is 00097c...
Proof:
Let X, Y and f be given.
Assume H1.
Apply H1 to the current goal.
Assume H1 _.
An exact proof term for the current goal is H1.
∎
Theorem. (bij_id)
∀X, bij X X (λx ⇒ x)
In Proofgold the corresponding term root is a8cefb... and proposition id is 707f37...
Proof:
Let X be given.
We will prove (∀u ∈ X, u ∈ X) ∧ (∀u v ∈ X, u = v → u = v) ∧ (∀w ∈ X, ∃u ∈ X, u = w).
Apply and3I to the current goal.
An exact proof term for the current goal is (λu Hu ⇒ Hu).
An exact proof term for the current goal is (λu Hu v Hv H1 ⇒ H1).
Let w be given.
Assume Hw.
We use w to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is Hw.
Use reflexivity.
∎
Theorem. (bij_comp)
∀X Y Z : set, ∀f g : set → set, bij X Y f → bij Y Z g → bij X Z (λx ⇒ g (f x))
In Proofgold the corresponding term root is abf851... and proposition id is 4c9045...
Proof:
Let X, Y, Z, f and g be given.
Assume Hf.
Apply Hf to the current goal.
Assume Hf12 Hf3.
Apply Hf12 to the current goal.
Assume Hf1 Hf2.
Assume Hg.
Apply Hg to the current goal.
Assume Hg12 Hg3.
Apply Hg12 to the current goal.
Assume Hg1 Hg2.
We will prove (∀u ∈ X, g (f u) ∈ Z) ∧ (∀u v ∈ X, g (f u) = g (f v) → u = v) ∧ (∀w ∈ Z, ∃u ∈ X, g (f u) = w).
Apply and3I to the current goal.
Let u be given.
Assume Hu: u ∈ X.
An exact proof term for the current goal is (Hg1 (f u) (Hf1 u Hu)).
Let u be given.
Assume Hu: u ∈ X.
Let v be given.
Assume Hv: v ∈ X.
Assume H1: g (f u) = g (f v).
We will prove u = v.
Apply Hf2 u Hu v Hv to the current goal.
We will prove f u = f v.
Apply Hg2 (f u) (Hf1 u Hu) (f v) (Hf1 v Hv) to the current goal.
We will prove g (f u) = g (f v).
An exact proof term for the current goal is H1.
Let w be given.
Assume Hw: w ∈ Z.
Apply Hg3 w Hw to the current goal.
Let y be given.
Assume Hy12.
Apply Hy12 to the current goal.
Assume Hy1: y ∈ Y.
Assume Hy2: g y = w.
Apply Hf3 y Hy1 to the current goal.
Let u be given.
Assume Hu12.
Apply Hu12 to the current goal.
Assume Hu1: u ∈ X.
Assume Hu2: f u = y.
We will prove ∃u ∈ X, g (f u) = w.
We use u to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is Hu1.
rewrite the current goal using Hu2 (from left to right).
An exact proof term for the current goal is Hy2.
∎
Theorem. (bij_surj)
∀X Y, ∀f : set → set, bij X Y f → surj X Y f
In Proofgold the corresponding term root is e7a049... and proposition id is 7fa9a5...
Proof:
Let X, Y and f be given.
Assume H1.
Apply H1 to the current goal.
Assume H1.
Apply H1 to the current goal.
Assume H1 _ H2.
We will prove (∀u ∈ X, f u ∈ Y) ∧ (∀w ∈ Y, ∃u ∈ X, f u = w).
Apply andI to the current goal.
An exact proof term for the current goal is H1.
An exact proof term for the current goal is H2.
∎
Definition. We define inv to be λX f ⇒ λy : set ⇒ Eps_i (λx ⇒ x ∈ X ∧ f x = y) of type set → (set → set) → set → set.
In Proofgold the corresponding term root is 896fa9... and object id is f3a015...
Theorem. (surj_rinv)
∀X Y, ∀f : set → set, (∀w ∈ Y, ∃u ∈ X, f u = w) → ∀y ∈ Y, inv X f y ∈ X ∧ f (inv X f y) = y
In Proofgold the corresponding term root is ebf371... and proposition id is 6489d2...
Proof:
Let X, Y and f be given.
Assume H1.
Let y be given.
Assume Hy: y ∈ Y.
Apply H1 y Hy to the current goal.
Let x be given.
Assume H2.
An exact proof term for the current goal is Eps_i_ax (λx ⇒ x ∈ X ∧ f x = y) x H2.
∎
Theorem. (inj_linv)
∀X, ∀f : set → set, (∀u v ∈ X, f u = f v → u = v) → ∀x ∈ X, inv X f (f x) = x
In Proofgold the corresponding term root is 18ecce... and proposition id is d16557...
Proof:
Let X and f be given.
Assume H1.
Let x be given.
Assume Hx.
We prove the intermediate claim L1: inv X f (f x) ∈ X ∧ f (inv X f (f x)) = f x.
Apply Eps_i_ax (λx' ⇒ x' ∈ X ∧ f x' = f x) x to the current goal.
Apply andI to the current goal.
An exact proof term for the current goal is Hx.
Use reflexivity.
Apply L1 to the current goal.
Assume H2 H3.
An exact proof term for the current goal is H1 (inv X f (f x)) H2 x Hx H3.
∎
Theorem. (bij_inv)
∀X Y, ∀f : set → set, bij X Y f → bij Y X (inv X f)
In Proofgold the corresponding term root is ff1332... and proposition id is 69c537...
Proof:
Let X, Y and f be given.
Assume H1.
Apply H1 to the current goal.
Assume H2.
Apply H2 to the current goal.
Assume H3: ∀u ∈ X, f u ∈ Y.
Assume H4: ∀u v ∈ X, f u = f v → u = v.
Assume H5: ∀w ∈ Y, ∃u ∈ X, f u = w.
Set g to be the term λy ⇒ Eps_i (λx ⇒ x ∈ X ∧ f x = y) of type set → set.
We prove the intermediate claim L1: ∀y ∈ Y, g y ∈ X ∧ f (g y) = y.
An exact proof term for the current goal is surj_rinv X Y f H5.
We will prove (∀u ∈ Y, g u ∈ X) ∧ (∀u v ∈ Y, g u = g v → u = v) ∧ (∀w ∈ X, ∃u ∈ Y, g u = w).
Apply and3I to the current goal.
We will prove ∀u ∈ Y, g u ∈ X.
Let u be given.
Assume Hu.
Apply L1 u Hu to the current goal.
Assume H _.
An exact proof term for the current goal is H.
We will prove ∀u v ∈ Y, g u = g v → u = v.
Let u be given.
Assume Hu.
Let v be given.
Assume Hv H6.
We will prove u = v.
Apply L1 u Hu to the current goal.
Assume H7: g u ∈ X.
Assume H8: f (g u) = u.
Apply L1 v Hv to the current goal.
Assume H9: g v ∈ X.
Assume H10: f (g v) = v.
rewrite the current goal using H8 (from right to left).
rewrite the current goal using H10 (from right to left).
rewrite the current goal using H6 (from right to left).
Use reflexivity.
We will prove ∀w ∈ X, ∃u ∈ Y, g u = w.
Let w be given.
Assume Hw.
We prove the intermediate claim Lfw: f w ∈ Y.
An exact proof term for the current goal is H3 w Hw.
We use f w to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is Lfw.
An exact proof term for the current goal is inj_linv X f H4 w Hw.
∎
Definition. We define atleastp to be λX Y : set ⇒ ∃f : set → set, inj X Y f of type set → set → prop.
In Proofgold the corresponding term root is 9bb9c2... and object id is 13545c...
Theorem. (atleastp_tra)
∀X Y Z, atleastp X Y → atleastp Y Z → atleastp X Z
In Proofgold the corresponding term root is 0e83bf... and proposition id is c412bf...
Proof:
The rest of this subproof is missing.
∎
Theorem. (Subq_atleastp)
∀X Y, X ⊆ Y → atleastp X Y
In Proofgold the corresponding term root is 507ac6... and proposition id is 3a427f...
Proof:
The rest of this subproof is missing.
∎
Definition. We define equip to be λX Y : set ⇒ ∃f : set → set, bij X Y f of type set → set → prop.
In Proofgold the corresponding term root is eb4419... and object id is 7f2bf6...
Theorem. (equip_atleastp)
∀X Y, equip X Y → atleastp X Y
In Proofgold the corresponding term root is c9d4bd... and proposition id is 9a412a...
Proof:
The rest of this subproof is missing.
∎
Theorem. (equip_ref)
∀X, equip X X
In Proofgold the corresponding term root is cc7e80... and proposition id is c41fb1...
Proof:
The rest of this subproof is missing.
∎
Theorem. (equip_sym)
∀X Y, equip X Y → equip Y X
In Proofgold the corresponding term root is f28954... and proposition id is 272f04...
Proof:
The rest of this subproof is missing.
∎
Theorem. (equip_tra)
∀X Y Z, equip X Y → equip Y Z → equip X Z
In Proofgold the corresponding term root is 322bfb... and proposition id is d616c1...
Proof:
The rest of this subproof is missing.
∎
Theorem. (equip_0_Empty)
∀X, equip X 0 → X = 0
In Proofgold the corresponding term root is 83ab2f... and proposition id is 4c101c...
Proof:
The rest of this subproof is missing.
∎
Theorem. (equip_adjoin_ordsucc)
∀N X y, y ∉ X → equip N X → equip (ordsucc N) (X ∪ {y})
In Proofgold the corresponding term root is eab190... and proposition id is e8bc03...
Proof:
The rest of this subproof is missing.
∎
Theorem. (equip_ordsucc_remove1)
∀X N, ∀x ∈ X, equip X (ordsucc N) → equip (X ∖ {x}) N
In Proofgold the corresponding term root is 997b32... and proposition id is 9c2236...
Proof:
The rest of this subproof is missing.
∎
Beginning of Section SchroederBernstein
Theorem. (KnasterTarski_set)
∀A, ∀F : set → set, (∀U ∈ 𝒫 A, F U ∈ 𝒫 A) → (∀U V ∈ 𝒫 A, U ⊆ V → F U ⊆ F V) → ∃Y ∈ 𝒫 A, F Y = Y
In Proofgold the corresponding term root is 2ffb9f... and proposition id is 31ef9d...
Proof:
The rest of this subproof is missing.
∎
Theorem. (image_In_Power)
∀A B, ∀f : set → set, (∀x ∈ A, f x ∈ B) → ∀U ∈ 𝒫 A, {f x|x ∈ U} ∈ 𝒫 B
In Proofgold the corresponding term root is 6799e8... and proposition id is 43ed06...
Proof:
The rest of this subproof is missing.
∎
Theorem. (image_monotone)
∀f : set → set, ∀U V, U ⊆ V → {f x|x ∈ U} ⊆ {f x|x ∈ V}
In Proofgold the corresponding term root is ac6fe8... and proposition id is f923c6...
Proof:
The rest of this subproof is missing.
∎
Theorem. (setminus_antimonotone)
∀A U V, U ⊆ V → A ∖ V ⊆ A ∖ U
In Proofgold the corresponding term root is 67bd0e... and proposition id is f9b735...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SchroederBernstein)
∀A B, ∀f g : set → set, inj A B f → inj B A g → equip A B
In Proofgold the corresponding term root is e247a8... and proposition id is 03a2cd...
Proof:
The rest of this subproof is missing.
∎
Theorem. (atleastp_antisym_equip)
∀A B, atleastp A B → atleastp B A → equip A B
In Proofgold the corresponding term root is e218ed... and proposition id is 2c48a8...
Proof:
The rest of this subproof is missing.
∎
End of Section SchroederBernstein
Beginning of Section PigeonHole
Theorem. (PigeonHole_nat)
∀n, nat_p n → ∀f : set → set, (∀i ∈ ordsucc n, f i ∈ n) → ¬ (∀i j ∈ ordsucc n, f i = f j → i = j)
In Proofgold the corresponding term root is d756b7... and proposition id is 604a84...
Proof:
The rest of this subproof is missing.
∎
Theorem. (Pigeonhole_not_atleastp_ordsucc)
∀n, nat_p n → ¬ atleastp (ordsucc n) n
In Proofgold the corresponding term root is 8a6bdc... and proposition id is 4fb58b...
Proof:
The rest of this subproof is missing.
∎
End of Section PigeonHole
Theorem. (Union_ordsucc_eq)
∀n, nat_p n → ⋃ (ordsucc n) = n
In Proofgold the corresponding term root is 87308c... and proposition id is e8cdeb...
Proof:
The rest of this subproof is missing.
∎
Theorem. (cases_1)
∀i ∈ 1, ∀p : set → prop, p 0 → p i
In Proofgold the corresponding term root is 706605... and proposition id is c6f0c0...
Proof:
The rest of this subproof is missing.
∎
Theorem. (cases_2)
∀i ∈ 2, ∀p : set → prop, p 0 → p 1 → p i
In Proofgold the corresponding term root is 4693b2... and proposition id is 74563a...
Proof:
The rest of this subproof is missing.
∎
Theorem. (neq_0_1)
0 ≠ 1
In Proofgold the corresponding term root is 9630fd... and proposition id is 74867a...
Proof:
The rest of this subproof is missing.
∎
Theorem. (neq_1_0)
1 ≠ 0
In Proofgold the corresponding term root is 3a0478... and proposition id is 952fed...
Proof:
The rest of this subproof is missing.
∎
Theorem. (neq_0_2)
0 ≠ 2
In Proofgold the corresponding term root is 055580... and proposition id is 9c9a3f...
Proof:
The rest of this subproof is missing.
∎
Theorem. (neq_2_0)
2 ≠ 0
In Proofgold the corresponding term root is 4d54d8... and proposition id is 00b357...
Proof:
The rest of this subproof is missing.
∎
Definition. We define ordinal to be λalpha : set ⇒ TransSet alpha ∧ ∀beta ∈ alpha, TransSet beta of type set → prop.
In Proofgold the corresponding term root is ee28d9... and object id is 53ee8f...
Theorem. (ordinal_TransSet)
∀alpha : set, ordinal alpha → TransSet alpha
In Proofgold the corresponding term root is 0dbaa3... and proposition id is c9e708...
Proof:
The rest of this subproof is missing.
∎
Theorem. (ordinal_Empty)
ordinal Empty
In Proofgold the corresponding term root is ce6bea... and proposition id is 7d95d1...
Proof:
The rest of this subproof is missing.
∎
Theorem. (ordinal_Hered)
∀alpha : set, ordinal alpha → ∀beta ∈ alpha, ordinal beta
In Proofgold the corresponding term root is a7e78b... and proposition id is 43b0f3...
Proof:
The rest of this subproof is missing.
∎
Theorem. (TransSet_ordsucc)
∀X : set, TransSet X → TransSet (ordsucc X)
In Proofgold the corresponding term root is fc2157... and proposition id is ebf9e3...
Proof:
The rest of this subproof is missing.
∎
Theorem. (ordinal_ordsucc)
∀alpha : set, ordinal alpha → ordinal (ordsucc alpha)
In Proofgold the corresponding term root is 0ba97f... and proposition id is ab0f7f...
Proof:
The rest of this subproof is missing.
∎
Theorem. (nat_p_ordinal)
∀n : set, nat_p n → ordinal n
In Proofgold the corresponding term root is 274fc4... and proposition id is 47cd49...
Proof:
The rest of this subproof is missing.
∎
Theorem. (ordinal_1)
ordinal 1
In Proofgold the corresponding term root is c94d41... and proposition id is e325a5...
Proof:
The rest of this subproof is missing.
∎
Theorem. (ordinal_2)
ordinal 2
In Proofgold the corresponding term root is a57075... and proposition id is e64a0d...
Proof:
The rest of this subproof is missing.
∎
Theorem. (TransSet_ordsucc_In_Subq)
∀X : set, TransSet X → ∀x ∈ X, ordsucc x ⊆ X
In Proofgold the corresponding term root is d5d882... and proposition id is 3c2e45...
Proof:
The rest of this subproof is missing.
∎
Theorem. (ordinal_ordsucc_In_Subq)
∀alpha, ordinal alpha → ∀beta ∈ alpha, ordsucc beta ⊆ alpha
In Proofgold the corresponding term root is 57345e... and proposition id is b00304...
Proof:
The rest of this subproof is missing.
∎
Theorem. (ordinal_trichotomy_or)
∀alpha beta : set, ordinal alpha → ordinal beta → alpha ∈ beta ∨ alpha = beta ∨ beta ∈ alpha
In Proofgold the corresponding term root is 078a34... and proposition id is 21f2d6...
Proof:
The rest of this subproof is missing.
∎
Theorem. (ordinal_trichotomy_or_impred)
∀alpha beta : set, ordinal alpha → ordinal beta → ∀p : prop, (alpha ∈ beta → p) → (alpha = beta → p) → (beta ∈ alpha → p) → p
In Proofgold the corresponding term root is 7c0f0b... and proposition id is bb6d2d...
Proof:
The rest of this subproof is missing.
∎
Theorem. (ordinal_In_Or_Subq)
∀alpha beta, ordinal alpha → ordinal beta → alpha ∈ beta ∨ beta ⊆ alpha
In Proofgold the corresponding term root is 70fe0d... and proposition id is 7b99b0...
Proof:
The rest of this subproof is missing.
∎
Theorem. (ordinal_linear)
∀alpha beta, ordinal alpha → ordinal beta → alpha ⊆ beta ∨ beta ⊆ alpha
In Proofgold the corresponding term root is 3f30a5... and proposition id is 3b1fce...
Proof:
The rest of this subproof is missing.
∎
Theorem. (ordinal_ordsucc_In_eq)
∀alpha beta, ordinal alpha → beta ∈ alpha → ordsucc beta ∈ alpha ∨ alpha = ordsucc beta
In Proofgold the corresponding term root is 500378... and proposition id is 0c19e7...
Proof:
The rest of this subproof is missing.
∎
Theorem. (ordinal_lim_or_succ)
∀alpha, ordinal alpha → (∀beta ∈ alpha, ordsucc beta ∈ alpha) ∨ (∃beta ∈ alpha, alpha = ordsucc beta)
In Proofgold the corresponding term root is 4450c0... and proposition id is a12697...
Proof:
The rest of this subproof is missing.
∎
Theorem. (ordinal_ordsucc_In)
∀alpha, ordinal alpha → ∀beta ∈ alpha, ordsucc beta ∈ ordsucc alpha
In Proofgold the corresponding term root is f4b1ed... and proposition id is 8ab1dc...
Proof:
The rest of this subproof is missing.
∎
Theorem. (ordinal_famunion)
∀X, ∀F : set → set, (∀x ∈ X, ordinal (F x)) → ordinal (⋃x ∈ XF x)
In Proofgold the corresponding term root is fc0359... and proposition id is b3f1e1...
Proof:
The rest of this subproof is missing.
∎
Theorem. (ordinal_binintersect)
∀alpha beta, ordinal alpha → ordinal beta → ordinal (alpha ∩ beta)
In Proofgold the corresponding term root is 43d507... and proposition id is 025328...
Proof:
The rest of this subproof is missing.
∎
Theorem. (ordinal_binunion)
∀alpha beta, ordinal alpha → ordinal beta → ordinal (alpha ∪ beta)
In Proofgold the corresponding term root is 861ae3... and proposition id is b24989...
Proof:
The rest of this subproof is missing.
∎
Theorem. (ordinal_ind)
∀p : set → prop, (∀alpha, ordinal alpha → (∀beta ∈ alpha, p beta) → p alpha) → ∀alpha, ordinal alpha → p alpha
In Proofgold the corresponding term root is bb5575... and proposition id is 973cd8...
Proof:
The rest of this subproof is missing.
∎
Theorem. (least_ordinal_ex)
∀p : set → prop, (∃alpha, ordinal alpha ∧ p alpha) → ∃alpha, ordinal alpha ∧ p alpha ∧ ∀beta ∈ alpha, ¬ p beta
In Proofgold the corresponding term root is df6c77... and proposition id is ab6fe1...
Proof:
The rest of this subproof is missing.
∎
Theorem. (equip_Sing_1)
∀x, equip {x} 1
In Proofgold the corresponding term root is be1ab2... and proposition id is 5169f4...
Proof:
The rest of this subproof is missing.
∎
Theorem. (TransSet_In_ordsucc_Subq)
∀x y, TransSet y → x ∈ ordsucc y → x ⊆ y
In Proofgold the corresponding term root is f80c6e... and proposition id is 341993...
Proof:
The rest of this subproof is missing.
∎
Theorem. (exandE_i)
∀P Q : set → prop, (∃x, P x ∧ Q x) → ∀r : prop, (∀x, P x → Q x → r) → r
In Proofgold the corresponding term root is 083f82... and proposition id is b1b26e...
Proof:
The rest of this subproof is missing.
∎
Theorem. (exandE_ii)
∀P Q : (set → set) → prop, (∃x : set → set, P x ∧ Q x) → ∀p : prop, (∀x : set → set, P x → Q x → p) → p
In Proofgold the corresponding term root is c3e995... and proposition id is ed9d65...
Proof:
The rest of this subproof is missing.
∎
Theorem. (exandE_iii)
∀P Q : (set → set → set) → prop, (∃x : set → set → set, P x ∧ Q x) → ∀p : prop, (∀x : set → set → set, P x → Q x → p) → p
In Proofgold the corresponding term root is 028a8f... and proposition id is 80b3b0...
Proof:
The rest of this subproof is missing.
∎
Theorem. (exandE_iiii)
∀P Q : (set → set → set → set) → prop, (∃x : set → set → set → set, P x ∧ Q x) → ∀p : prop, (∀x : set → set → set → set, P x → Q x → p) → p
In Proofgold the corresponding term root is 7807d4... and proposition id is d1536a...
Proof:
The rest of this subproof is missing.
∎
Beginning of Section Descr_ii
Variable P : (set → set) → prop
Definition. We define Descr_ii to be λx : set ⇒ Eps_i (λy ⇒ ∀h : set → set, P h → h x = y) of type set → set.
In Proofgold the corresponding term root is 3bae39... and object id is 3a4ad2...
Hypothesis Pex : ∃f : set → set, P f
Hypothesis Puniq : ∀f g : set → set, P f → P g → f = g
Theorem. (Descr_ii_prop)
P Descr_ii
In Proofgold the corresponding term root is 45bfc0... and proposition id is e705d1...
Proof:
The rest of this subproof is missing.
∎
End of Section Descr_ii
Beginning of Section Descr_iii
Variable P : (set → set → set) → prop
Definition. We define Descr_iii to be λx y : set ⇒ Eps_i (λz ⇒ ∀h : set → set → set, P h → h x y = z) of type set → set → set.
In Proofgold the corresponding term root is ca5fc1... and object id is 877e40...
Hypothesis Pex : ∃f : set → set → set, P f
Hypothesis Puniq : ∀f g : set → set → set, P f → P g → f = g
Theorem. (Descr_iii_prop)
P Descr_iii
In Proofgold the corresponding term root is 261c3d... and proposition id is 160421...
Proof:
The rest of this subproof is missing.
∎
End of Section Descr_iii
Beginning of Section Descr_Vo1
Variable P : Vo 1 → prop
Definition. We define Descr_Vo1 to be λx : set ⇒ ∀h : set → prop, P h → h x of type Vo 1.
In Proofgold the corresponding term root is 615c0a... and object id is a4ecdd...
Hypothesis Pex : ∃f : Vo 1, P f
Hypothesis Puniq : ∀f g : Vo 1, P f → P g → f = g
Theorem. (Descr_Vo1_prop)
P Descr_Vo1
In Proofgold the corresponding term root is f550ef... and proposition id is 2213e2...
Proof:
The rest of this subproof is missing.
∎
End of Section Descr_Vo1
Beginning of Section If_ii
Variable p : prop
Variable f g : set → set
Definition. We define If_ii to be λx ⇒ if p then f x else g x of type set → set.
In Proofgold the corresponding term root is 17057f... and object id is e29015...
Theorem. (If_ii_1)
p → If_ii = f
In Proofgold the corresponding term root is 31521e... and proposition id is 342a1f...
Proof:
The rest of this subproof is missing.
∎
Theorem. (If_ii_0)
¬ p → If_ii = g
In Proofgold the corresponding term root is 6acab0... and proposition id is 36418b...
Proof:
The rest of this subproof is missing.
∎
End of Section If_ii
Beginning of Section If_iii
Variable p : prop
Variable f g : set → set → set
Definition. We define If_iii to be λx y ⇒ if p then f x y else g x y of type set → set → set.
In Proofgold the corresponding term root is 331476... and object id is dd7e2b...
Theorem. (If_iii_1)
p → If_iii = f
In Proofgold the corresponding term root is 2d62ee... and proposition id is 48bc10...
Proof:
The rest of this subproof is missing.
∎
Theorem. (If_iii_0)
¬ p → If_iii = g
In Proofgold the corresponding term root is 9b77d9... and proposition id is d11201...
Proof:
The rest of this subproof is missing.
∎
End of Section If_iii
Beginning of Section EpsilonRec_i
Variable F : set → (set → set) → set
Definition. We define In_rec_i_G to be λX Y ⇒ ∀R : set → set → prop, (∀X : set, ∀f : set → set, (∀x ∈ X, R x (f x)) → R X (F X f)) → R X Y of type set → set → prop.
In Proofgold the corresponding term root is fde509... and object id is a81e18...
Definition. We define In_rec_i to be λX ⇒ Eps_i (In_rec_i_G X) of type set → set.
In Proofgold the corresponding term root is fac413... and object id is 93aac4...
Theorem. (In_rec_i_G_c)
∀X : set, ∀f : set → set, (∀x ∈ X, In_rec_i_G x (f x)) → In_rec_i_G X (F X f)
In Proofgold the corresponding term root is 4da597... and proposition id is bc452e...
Proof:
The rest of this subproof is missing.
∎
Theorem. (In_rec_i_G_inv)
∀X : set, ∀Y : set, In_rec_i_G X Y → ∃f : set → set, (∀x ∈ X, In_rec_i_G x (f x)) ∧ Y = F X f
In Proofgold the corresponding term root is ba1774... and proposition id is 5cfac2...
Proof:
The rest of this subproof is missing.
∎
Hypothesis Fr : ∀X : set, ∀g h : set → set, (∀x ∈ X, g x = h x) → F X g = F X h
Theorem. (In_rec_i_G_f)
∀X : set, ∀Y Z : set, In_rec_i_G X Y → In_rec_i_G X Z → Y = Z
In Proofgold the corresponding term root is 25f7ff... and proposition id is 02b4e4...
Proof:
The rest of this subproof is missing.
∎
Theorem. (In_rec_i_G_In_rec_i)
∀X : set, In_rec_i_G X (In_rec_i X)
In Proofgold the corresponding term root is 095407... and proposition id is dc2359...
Proof:
The rest of this subproof is missing.
∎
Theorem. (In_rec_i_G_In_rec_i_d)
∀X : set, In_rec_i_G X (F X In_rec_i)
In Proofgold the corresponding term root is b480fe... and proposition id is 5b92ff...
Proof:
The rest of this subproof is missing.
∎
Theorem. (In_rec_i_eq)
∀X : set, In_rec_i X = F X In_rec_i
In Proofgold the corresponding term root is 110653... and proposition id is e8653b...
Proof:
The rest of this subproof is missing.
∎
End of Section EpsilonRec_i
Beginning of Section EpsilonRec_ii
Variable F : set → (set → (set → set)) → (set → set)
Definition. We define In_rec_G_ii to be λX Y ⇒ ∀R : set → (set → set) → prop, (∀X : set, ∀f : set → (set → set), (∀x ∈ X, R x (f x)) → R X (F X f)) → R X Y of type set → (set → set) → prop.
In Proofgold the corresponding term root is 6edba9... and object id is b31dbf...
Definition. We define In_rec_ii to be λX ⇒ Descr_ii (In_rec_G_ii X) of type set → (set → set).
In Proofgold the corresponding term root is f3c9ab... and object id is 3841bd...
Theorem. (In_rec_G_ii_c)
∀X : set, ∀f : set → (set → set), (∀x ∈ X, In_rec_G_ii x (f x)) → In_rec_G_ii X (F X f)
In Proofgold the corresponding term root is 381613... and proposition id is 49fbeb...
Proof:
The rest of this subproof is missing.
∎
Theorem. (In_rec_G_ii_inv)
∀X : set, ∀Y : (set → set), In_rec_G_ii X Y → ∃f : set → (set → set), (∀x ∈ X, In_rec_G_ii x (f x)) ∧ Y = F X f
In Proofgold the corresponding term root is 336cfe... and proposition id is b4a180...
Proof:
The rest of this subproof is missing.
∎
Hypothesis Fr : ∀X : set, ∀g h : set → (set → set), (∀x ∈ X, g x = h x) → F X g = F X h
Theorem. (In_rec_G_ii_f)
∀X : set, ∀Y Z : (set → set), In_rec_G_ii X Y → In_rec_G_ii X Z → Y = Z
In Proofgold the corresponding term root is 64b6e8... and proposition id is a3826b...
Proof:
The rest of this subproof is missing.
∎
Theorem. (In_rec_G_ii_In_rec_ii)
∀X : set, In_rec_G_ii X (In_rec_ii X)
In Proofgold the corresponding term root is b2f81f... and proposition id is 2893f4...
Proof:
The rest of this subproof is missing.
∎
Theorem. (In_rec_G_ii_In_rec_ii_d)
∀X : set, In_rec_G_ii X (F X In_rec_ii)
In Proofgold the corresponding term root is 2ff1ca... and proposition id is f12072...
Proof:
The rest of this subproof is missing.
∎
Theorem. (In_rec_ii_eq)
∀X : set, In_rec_ii X = F X In_rec_ii
In Proofgold the corresponding term root is f7aff6... and proposition id is 7bb1e9...
Proof:
The rest of this subproof is missing.
∎
End of Section EpsilonRec_ii
Beginning of Section EpsilonRec_iii
Variable F : set → (set → (set → set → set)) → (set → set → set)
Definition. We define In_rec_G_iii to be λX Y ⇒ ∀R : set → (set → set → set) → prop, (∀X : set, ∀f : set → (set → set → set), (∀x ∈ X, R x (f x)) → R X (F X f)) → R X Y of type set → (set → set → set) → prop.
In Proofgold the corresponding term root is 533cc7... and object id is 7a2f6b...
Definition. We define In_rec_iii to be λX ⇒ Descr_iii (In_rec_G_iii X) of type set → (set → set → set).
In Proofgold the corresponding term root is 9b3a85... and object id is 651077...
Theorem. (In_rec_G_iii_c)
∀X : set, ∀f : set → (set → set → set), (∀x ∈ X, In_rec_G_iii x (f x)) → In_rec_G_iii X (F X f)
In Proofgold the corresponding term root is 4ea572... and proposition id is a7d90f...
Proof:
The rest of this subproof is missing.
∎
Theorem. (In_rec_G_iii_inv)
∀X : set, ∀Y : (set → set → set), In_rec_G_iii X Y → ∃f : set → (set → set → set), (∀x ∈ X, In_rec_G_iii x (f x)) ∧ Y = F X f
In Proofgold the corresponding term root is 5f6e0c... and proposition id is f47dae...
Proof:
The rest of this subproof is missing.
∎
Hypothesis Fr : ∀X : set, ∀g h : set → (set → set → set), (∀x ∈ X, g x = h x) → F X g = F X h
Theorem. (In_rec_G_iii_f)
∀X : set, ∀Y Z : (set → set → set), In_rec_G_iii X Y → In_rec_G_iii X Z → Y = Z
In Proofgold the corresponding term root is 574d71... and proposition id is 40978a...
Proof:
The rest of this subproof is missing.
∎
Theorem. (In_rec_G_iii_In_rec_iii)
∀X : set, In_rec_G_iii X (In_rec_iii X)
In Proofgold the corresponding term root is d8c3ba... and proposition id is 37f9a7...
Proof:
The rest of this subproof is missing.
∎
Theorem. (In_rec_G_iii_In_rec_iii_d)
∀X : set, In_rec_G_iii X (F X In_rec_iii)
In Proofgold the corresponding term root is 60c6d7... and proposition id is c05f77...
Proof:
The rest of this subproof is missing.
∎
Theorem. (In_rec_iii_eq)
∀X : set, In_rec_iii X = F X In_rec_iii
In Proofgold the corresponding term root is 5f0574... and proposition id is a0709d...
Proof:
The rest of this subproof is missing.
∎
End of Section EpsilonRec_iii
Beginning of Section NatRec
Variable z : set
Variable f : set → set → set
Let F : set → (set → set) → set ≝ λn g ⇒ if ⋃ n ∈ n then f (⋃ n) (g (⋃ n)) else z
Definition. We define nat_primrec to be In_rec_i F of type set → set.
In Proofgold the corresponding term root is 3be1c5... and object id is 1a64d5...
Theorem. (nat_primrec_r)
∀X : set, ∀g h : set → set, (∀x ∈ X, g x = h x) → F X g = F X h
In Proofgold the corresponding term root is c955cd... and proposition id is 9ba887...
Proof:
The rest of this subproof is missing.
∎
Theorem. (nat_primrec_0)
nat_primrec 0 = z
In Proofgold the corresponding term root is 416eaf... and proposition id is be11a2...
Proof:
The rest of this subproof is missing.
∎
Theorem. (nat_primrec_S)
∀n : set, nat_p n → nat_primrec (ordsucc n) = f n (nat_primrec n)
In Proofgold the corresponding term root is 797185... and proposition id is fe4c88...
Proof:
The rest of this subproof is missing.
∎
End of Section NatRec
Beginning of Section NatAdd
Definition. We define add_nat to be λn m : set ⇒ nat_primrec n (λ_ r ⇒ ordsucc r) m of type set → set → set.
In Proofgold the corresponding term root is afa8ae... and object id is 29768b...
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_nat.
Theorem. (add_nat_0R)
∀n : set, n + 0 = n
In Proofgold the corresponding term root is 402bcb... and proposition id is 23277f...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_nat_SR)
∀n m : set, nat_p m → n + ordsucc m = ordsucc (n + m)
In Proofgold the corresponding term root is a53523... and proposition id is e61e18...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_nat_p)
∀n : set, nat_p n → ∀m : set, nat_p m → nat_p (n + m)
In Proofgold the corresponding term root is a68eaa... and proposition id is 4c1d5d...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_nat_1_1_2)
1 + 1 = 2
In Proofgold the corresponding term root is 174b87... and proposition id is 77f23f...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_nat_asso)
∀n : set, nat_p n → ∀m : set, nat_p m → ∀k : set, nat_p k → (n + m) + k = n + (m + k)
In Proofgold the corresponding term root is c81c09... and proposition id is eec294...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_nat_0L)
∀m : set, nat_p m → 0 + m = m
In Proofgold the corresponding term root is 8ceeb9... and proposition id is 84fbe4...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_nat_SL)
∀n : set, nat_p n → ∀m : set, nat_p m → ordsucc n + m = ordsucc (n + m)
In Proofgold the corresponding term root is 9c442a... and proposition id is e6efb7...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_nat_com)
∀n : set, nat_p n → ∀m : set, nat_p m → n + m = m + n
In Proofgold the corresponding term root is 40e952... and proposition id is 4ab179...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_nat_In_R)
∀m, nat_p m → ∀k ∈ m, ∀n, nat_p n → k + n ∈ m + n
In Proofgold the corresponding term root is eeaa55... and proposition id is 85b3ac...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_nat_In_L)
∀n, nat_p n → ∀m, nat_p m → ∀k ∈ m, n + k ∈ n + m
In Proofgold the corresponding term root is 0572fd... and proposition id is eaa26c...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_nat_Subq_R)
∀k, nat_p k → ∀m, nat_p m → k ⊆ m → ∀n, nat_p n → k + n ⊆ m + n
In Proofgold the corresponding term root is 0d6a07... and proposition id is d59a02...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_nat_Subq_L)
∀n, nat_p n → ∀k, nat_p k → ∀m, nat_p m → k ⊆ m → n + k ⊆ n + m
In Proofgold the corresponding term root is b7d52c... and proposition id is 586cfc...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_nat_Subq_R')
∀m, nat_p m → ∀n, nat_p n → m ⊆ m + n
In Proofgold the corresponding term root is ed3e7e... and proposition id is cff680...
Proof:
The rest of this subproof is missing.
∎
Theorem. (nat_Subq_add_ex)
∀n, nat_p n → ∀m, nat_p m → n ⊆ m → ∃k, nat_p k ∧ m = k + n
In Proofgold the corresponding term root is a1e406... and proposition id is 2c7fa4...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_nat_cancel_R)
∀k, nat_p k → ∀m, nat_p m → ∀n, nat_p n → k + n = m + n → k = m
In Proofgold the corresponding term root is acfe90... and proposition id is 1c3ee5...
Proof:
The rest of this subproof is missing.
∎
End of Section NatAdd
Beginning of Section NatMul
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_nat.
Definition. We define mul_nat to be λn m : set ⇒ nat_primrec 0 (λ_ r ⇒ n + r) m of type set → set → set.
In Proofgold the corresponding term root is 35a1ef... and object id is 80ccc6...
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_nat.
Theorem. (mul_nat_0R)
∀n : set, n * 0 = 0
In Proofgold the corresponding term root is b2f56c... and proposition id is 1e5b2c...
Proof:
The rest of this subproof is missing.
∎
Theorem. (mul_nat_SR)
∀n m, nat_p m → n * ordsucc m = n + n * m
In Proofgold the corresponding term root is 37df54... and proposition id is 6dd6fc...
Proof:
The rest of this subproof is missing.
∎
Theorem. (mul_nat_1R)
∀n, n * 1 = n
In Proofgold the corresponding term root is cab8eb... and proposition id is a3051d...
Proof:
The rest of this subproof is missing.
∎
Theorem. (mul_nat_p)
∀n : set, nat_p n → ∀m : set, nat_p m → nat_p (n * m)
In Proofgold the corresponding term root is d63438... and proposition id is e910cb...
Proof:
The rest of this subproof is missing.
∎
Theorem. (mul_nat_0L)
∀m : set, nat_p m → 0 * m = 0
In Proofgold the corresponding term root is db8a35... and proposition id is 6a2dd6...
Proof:
The rest of this subproof is missing.
∎
Theorem. (mul_nat_SL)
∀n : set, nat_p n → ∀m : set, nat_p m → ordsucc n * m = n * m + m
In Proofgold the corresponding term root is 5cf7ec... and proposition id is 24e4eb...
Proof:
The rest of this subproof is missing.
∎
Theorem. (mul_nat_com)
∀n : set, nat_p n → ∀m : set, nat_p m → n * m = m * n
In Proofgold the corresponding term root is 4525ef... and proposition id is 5d6ce4...
Proof:
The rest of this subproof is missing.
∎
Theorem. (mul_add_nat_distrL)
∀n : set, nat_p n → ∀m : set, nat_p m → ∀k : set, nat_p k → n * (m + k) = n * m + n * k
In Proofgold the corresponding term root is 7758f1... and proposition id is d57b02...
Proof:
The rest of this subproof is missing.
∎
Theorem. (mul_nat_asso)
∀n : set, nat_p n → ∀m : set, nat_p m → ∀k : set, nat_p k → (n * m) * k = n * (m * k)
In Proofgold the corresponding term root is 564886... and proposition id is 229d2c...
Proof:
The rest of this subproof is missing.
∎
Theorem. (mul_nat_Subq_R)
∀m n, nat_p m → nat_p n → m ⊆ n → ∀k, nat_p k → m * k ⊆ n * k
In Proofgold the corresponding term root is d99b10... and proposition id is 4324d1...
Proof:
The rest of this subproof is missing.
∎
Theorem. (mul_nat_Subq_L)
∀m n, nat_p m → nat_p n → m ⊆ n → ∀k, nat_p k → k * m ⊆ k * n
In Proofgold the corresponding term root is 929864... and proposition id is df9ccb...
Proof:
The rest of this subproof is missing.
∎
Theorem. (mul_nat_0_or_Subq)
∀m, nat_p m → ∀n, nat_p n → n = 0 ∨ m ⊆ m * n
In Proofgold the corresponding term root is bdf0eb... and proposition id is 2bbf0e...
Proof:
The rest of this subproof is missing.
∎
Theorem. (mul_nat_0_inv)
∀m, nat_p m → ∀n, nat_p n → m * n = 0 → m = 0 ∨ n = 0
In Proofgold the corresponding term root is 2da221... and proposition id is f3fbb4...
Proof:
The rest of this subproof is missing.
∎
Theorem. (mul_nat_0m_1n_In)
∀m, nat_p m → ∀n, nat_p n → 0 ∈ m → 1 ∈ n → m ∈ m * n
In Proofgold the corresponding term root is 611f85... and proposition id is 5d96a9...
Proof:
The rest of this subproof is missing.
∎
Theorem. (nat_le1_cases)
∀m, nat_p m → m ⊆ 1 → m = 0 ∨ m = 1
In Proofgold the corresponding term root is 591dbf... and proposition id is 5d5690...
Proof:
The rest of this subproof is missing.
∎
Definition. We define Pi_nat to be λf n ⇒ nat_primrec 1 (λi r ⇒ r * f i) n of type (set → set) → set → set.
In Proofgold the corresponding term root is a0b865... and object id is 70d0e8...
Theorem. (Pi_nat_0)
∀f : set → set, Pi_nat f 0 = 1
In Proofgold the corresponding term root is 6af04c... and proposition id is b7ebea...
Proof:
The rest of this subproof is missing.
∎
Theorem. (Pi_nat_S)
∀f : set → set, ∀n, nat_p n → Pi_nat f (ordsucc n) = Pi_nat f n * f n
In Proofgold the corresponding term root is 15f21a... and proposition id is b5fbbf...
Proof:
The rest of this subproof is missing.
∎
Theorem. (Pi_nat_p)
∀f : set → set, ∀n, nat_p n → (∀i ∈ n, nat_p (f i)) → nat_p (Pi_nat f n)
In Proofgold the corresponding term root is 1e8072... and proposition id is 58005c...
Proof:
The rest of this subproof is missing.
∎
Theorem. (Pi_nat_0_inv)
∀f : set → set, ∀n, nat_p n → (∀i ∈ n, nat_p (f i)) → Pi_nat f n = 0 → (∃i ∈ n, f i = 0)
In Proofgold the corresponding term root is 688c96... and proposition id is 98d7e0...
Proof:
The rest of this subproof is missing.
∎
Definition. We define exp_nat to be λn m : set ⇒ nat_primrec 1 (λ_ r ⇒ n * r) m of type set → set → set.
In Proofgold the corresponding term root is 37c531... and object id is 8e35a1...
Notation. We use ^ as an infix operator with priority 342 and which associates to the right corresponding to applying term exp_nat.
Theorem. (exp_nat_0)
∀n, n ^ 0 = 1
In Proofgold the corresponding term root is e32008... and proposition id is 856b46...
Proof:
The rest of this subproof is missing.
∎
Theorem. (exp_nat_S)
∀n m, nat_p m → n ^ (ordsucc m) = n * n ^ m
In Proofgold the corresponding term root is e9c4ce... and proposition id is caaf40...
Proof:
The rest of this subproof is missing.
∎
Theorem. (exp_nat_p)
∀n, nat_p n → ∀m, nat_p m → nat_p (n ^ m)
In Proofgold the corresponding term root is 1b73e0... and proposition id is 4f4027...
Proof:
The rest of this subproof is missing.
∎
Theorem. (exp_nat_1)
∀n, n ^ 1 = n
In Proofgold the corresponding term root is 5858d8... and proposition id is 3e9f77...
Proof:
The rest of this subproof is missing.
∎
End of Section NatMul
Beginning of Section form100_52
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_nat.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_nat.
Notation. We use ^ as an infix operator with priority 342 and which associates to the right corresponding to applying term exp_nat.
Theorem. (Subq_Sing0_1)
{0} ⊆ 1
In Proofgold the corresponding term root is cd7864... and proposition id is 1806bf...
Proof:
The rest of this subproof is missing.
∎
Theorem. (Subq_1_Sing0)
1 ⊆ {0}
In Proofgold the corresponding term root is aeae9a... and proposition id is abab2d...
Proof:
The rest of this subproof is missing.
∎
Theorem. (eq_1_Sing0)
1 = {0}
In Proofgold the corresponding term root is 5e5996... and proposition id is 264f38...
Proof:
The rest of this subproof is missing.
∎
Theorem. (Power_0_Sing_0)
𝒫 0 = {0}
In Proofgold the corresponding term root is 6526a1... and proposition id is a39fd0...
Proof:
The rest of this subproof is missing.
∎
Theorem. (equip_finite_Power)
∀n, nat_p n → ∀X, equip X n → equip (𝒫 X) (2 ^ n)
In Proofgold the corresponding term root is 0058ef... and proposition id is b4d9c8...
Proof:
The rest of this subproof is missing.
∎
End of Section form100_52
Theorem. (ZF_closed_E)
∀U, ZF_closed U → ∀p : prop, (Union_closed U → Power_closed U → Repl_closed U → p) → p
In Proofgold the corresponding term root is d795d4... and proposition id is c78c79...
Proof:
The rest of this subproof is missing.
∎
Theorem. (ZF_Union_closed)
∀U, ZF_closed U → ∀X ∈ U, ⋃ X ∈ U
In Proofgold the corresponding term root is a7cdf1... and proposition id is 250496...
Proof:
The rest of this subproof is missing.
∎
Theorem. (ZF_Power_closed)
∀U, ZF_closed U → ∀X ∈ U, 𝒫 X ∈ U
In Proofgold the corresponding term root is 5d781f... and proposition id is 7411dd...
Proof:
The rest of this subproof is missing.
∎
Theorem. (ZF_Repl_closed)
∀U, ZF_closed U → ∀X ∈ U, ∀F : set → set, (∀x ∈ X, F x ∈ U) → {F x|x ∈ X} ∈ U
In Proofgold the corresponding term root is 312268... and proposition id is 18f8ac...
Proof:
The rest of this subproof is missing.
∎
Theorem. (ZF_UPair_closed)
∀U, ZF_closed U → ∀x y ∈ U, {x,y} ∈ U
In Proofgold the corresponding term root is 54256a... and proposition id is ef42d3...
Proof:
The rest of this subproof is missing.
∎
Theorem. (ZF_Sing_closed)
∀U, ZF_closed U → ∀x ∈ U, {x} ∈ U
In Proofgold the corresponding term root is 5d8a83... and proposition id is d02d66...
Proof:
The rest of this subproof is missing.
∎
Theorem. (ZF_binunion_closed)
∀U, ZF_closed U → ∀X Y ∈ U, (X ∪ Y) ∈ U
In Proofgold the corresponding term root is 8a6240... and proposition id is 1bacfd...
Proof:
The rest of this subproof is missing.
∎
Theorem. (ZF_ordsucc_closed)
∀U, ZF_closed U → ∀x ∈ U, ordsucc x ∈ U
In Proofgold the corresponding term root is 68a6ba... and proposition id is 946e61...
Proof:
The rest of this subproof is missing.
∎
Theorem. (nat_p_UnivOf_Empty)
∀n : set, nat_p n → n ∈ UnivOf Empty
In Proofgold the corresponding term root is c885b4... and proposition id is 645aee...
Proof:
The rest of this subproof is missing.
∎
Definition. We define ω to be {n ∈ UnivOf Empty|nat_p n} of type set.
In Proofgold the corresponding term root is 6fc30a... and object id is ad9ad8...
Theorem. (omega_nat_p)
∀n ∈ ω, nat_p n
In Proofgold the corresponding term root is f44a30... and proposition id is c2c61b...
Proof:
The rest of this subproof is missing.
∎
Theorem. (nat_p_omega)
∀n : set, nat_p n → n ∈ ω
In Proofgold the corresponding term root is 6a1b44... and proposition id is d5080f...
Proof:
The rest of this subproof is missing.
∎
Theorem. (omega_ordsucc)
∀n ∈ ω, ordsucc n ∈ ω
In Proofgold the corresponding term root is 246d63... and proposition id is 17fb64...
Proof:
The rest of this subproof is missing.
∎
Theorem. (form100_22_v2)
∀f : set → set, ¬ inj (𝒫 ω) ω f
In Proofgold the corresponding term root is eabc14... and proposition id is 5eb9bd...
Proof:
The rest of this subproof is missing.
∎
Theorem. (form100_22_v3)
∀g : set → set, ¬ surj ω (𝒫 ω) g
In Proofgold the corresponding term root is 4de6f8... and proposition id is b20e6a...
Proof:
The rest of this subproof is missing.
∎
Theorem. (form100_22_v1)
¬ equip ω (𝒫 ω)
In Proofgold the corresponding term root is 7ab652... and proposition id is a76888...
Proof:
The rest of this subproof is missing.
∎
Theorem. (omega_TransSet)
TransSet ω
In Proofgold the corresponding term root is 8e89e6... and proposition id is c79e75...
Proof:
The rest of this subproof is missing.
∎
Theorem. (omega_ordinal)
ordinal ω
In Proofgold the corresponding term root is 3d621a... and proposition id is 177484...
Proof:
The rest of this subproof is missing.
∎
Theorem. (ordsucc_omega_ordinal)
ordinal (ordsucc ω)
In Proofgold the corresponding term root is 66659c... and proposition id is ecfad7...
Proof:
The rest of this subproof is missing.
∎
Definition. We define finite to be λX ⇒ ∃n ∈ ω, equip X n of type set → prop.
In Proofgold the corresponding term root is 04632d... and object id is 4c54a5...
Theorem. (nat_finite)
∀n, nat_p n → finite n
In Proofgold the corresponding term root is a27dbe... and proposition id is 9b83bf...
Proof:
The rest of this subproof is missing.
∎
Theorem. (finite_ind)
∀p : set → prop, p Empty → (∀X y, finite X → y ∉ X → p X → p (X ∪ {y})) → ∀X, finite X → p X
In Proofgold the corresponding term root is c53cb1... and proposition id is 19baca...
Proof:
The rest of this subproof is missing.
∎
Theorem. (finite_Empty)
finite 0
In Proofgold the corresponding term root is c073c6... and proposition id is ec2c27...
Proof:
The rest of this subproof is missing.
∎
Theorem. (Sing_finite)
∀x, finite {x}
In Proofgold the corresponding term root is f7016a... and proposition id is 281486...
Proof:
The rest of this subproof is missing.
∎
Theorem. (adjoin_finite)
∀X y, finite X → finite (X ∪ {y})
In Proofgold the corresponding term root is def317... and proposition id is e6aa6d...
Proof:
The rest of this subproof is missing.
∎
Theorem. (binunion_finite)
∀X, finite X → ∀Y, finite Y → finite (X ∪ Y)
In Proofgold the corresponding term root is fc26bb... and proposition id is 3365a6...
Proof:
The rest of this subproof is missing.
∎
Theorem. (famunion_nat_finite)
∀X : set → set, ∀n, nat_p n → (∀i ∈ n, finite (X i)) → finite (⋃i ∈ nX i)
In Proofgold the corresponding term root is 7f709a... and proposition id is f84ed0...
Proof:
The rest of this subproof is missing.
∎
Theorem. (Subq_finite)
∀X, finite X → ∀Y, Y ⊆ X → finite Y
In Proofgold the corresponding term root is 6b062c... and proposition id is e4e3ee...
Proof:
The rest of this subproof is missing.
∎
Definition. We define infinite to be λA ⇒ ¬ finite A of type set → prop.
In Proofgold the corresponding term root is 313e7b... and object id is fad8ea...
Beginning of Section InfinitePrimes
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_nat.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_nat.
Definition. We define divides_nat to be λm n ⇒ m ∈ ω ∧ n ∈ ω ∧ ∃k ∈ ω, m * k = n of type set → set → prop.
In Proofgold the corresponding term root is 1b17d5... and object id is 355928...
Theorem. (divides_nat_ref)
∀n, nat_p n → divides_nat n n
In Proofgold the corresponding term root is 98e754... and proposition id is 913818...
Proof:
The rest of this subproof is missing.
∎
Theorem. (divides_nat_tra)
∀k m n, divides_nat k m → divides_nat m n → divides_nat k n
In Proofgold the corresponding term root is f799b9... and proposition id is e4e38e...
Proof:
The rest of this subproof is missing.
∎
Definition. We define prime_nat to be λn ⇒ n ∈ ω ∧ 1 ∈ n ∧ ∀k ∈ ω, divides_nat k n → k = 1 ∨ k = n of type set → prop.
In Proofgold the corresponding term root is 729ee8... and object id is a54ca4...
Theorem. (divides_nat_mulR)
∀m n ∈ ω, divides_nat m (m * n)
In Proofgold the corresponding term root is eec687... and proposition id is ceb239...
Proof:
The rest of this subproof is missing.
∎
Theorem. (divides_nat_mulL)
∀m n ∈ ω, divides_nat n (m * n)
In Proofgold the corresponding term root is cc5e38... and proposition id is d5d991...
Proof:
The rest of this subproof is missing.
∎
Theorem. (Pi_nat_divides)
∀f : set → set, ∀n, nat_p n → (∀i ∈ n, nat_p (f i)) → (∀i ∈ n, divides_nat (f i) (Pi_nat f n))
In Proofgold the corresponding term root is 5e46a8... and proposition id is 2325b8...
Proof:
The rest of this subproof is missing.
∎
Definition. We define composite_nat to be λn ⇒ n ∈ ω ∧ ∃k m ∈ ω, 1 ∈ k ∧ 1 ∈ m ∧ k * m = n of type set → prop.
In Proofgold the corresponding term root is 38d880... and object id is 3dafd8...
Theorem. (prime_nat_or_composite_nat)
∀n ∈ ω, 1 ∈ n → prime_nat n ∨ composite_nat n
In Proofgold the corresponding term root is a8970f... and proposition id is a1c7fc...
Proof:
The rest of this subproof is missing.
∎
Theorem. (prime_nat_divisor_ex)
∀n, nat_p n → 1 ∈ n → ∃p, prime_nat p ∧ divides_nat p n
In Proofgold the corresponding term root is 2a24d5... and proposition id is 979fd7...
Proof:
The rest of this subproof is missing.
∎
Theorem. (nat_1In_not_divides_ordsucc)
∀m n, 1 ∈ m → divides_nat m n → ¬ divides_nat m (ordsucc n)
In Proofgold the corresponding term root is 96fffb... and proposition id is 8d635d...
Proof:
The rest of this subproof is missing.
∎
Definition. We define primes to be {n ∈ ω|prime_nat n} of type set.
In Proofgold the corresponding term root is f3caf8... and object id is 98dde1...
Theorem. (form100_11_infinite_primes)
infinite primes
In Proofgold the corresponding term root is 34eb3c... and proposition id is 13f23d...
Proof:
The rest of this subproof is missing.
∎
End of Section InfinitePrimes
Beginning of Section InfiniteRamsey
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_nat.
Theorem. (atleastp_omega_infinite)
∀X, atleastp ω X → infinite X
In Proofgold the corresponding term root is 0b0e2a... and proposition id is c1147e...
Proof:
The rest of this subproof is missing.
∎
Theorem. (infinite_remove1)
∀X, infinite X → ∀y, infinite (X ∖ {y})
In Proofgold the corresponding term root is f236a9... and proposition id is 7a03dc...
Proof:
The rest of this subproof is missing.
∎
Theorem. (infinite_Finite_Subq_ex)
∀X, infinite X → ∀n, nat_p n → ∃Y ⊆ X, equip Y n
In Proofgold the corresponding term root is d43352... and proposition id is da00fc...
Proof:
The rest of this subproof is missing.
∎
Theorem. (infiniteRamsey_lem)
∀X, ∀f g f' : set → set, infinite X → (∀Z ⊆ X, infinite Z → f Z ⊆ Z ∧ infinite (f Z)) → (∀Z ⊆ X, infinite Z → g Z ∈ Z ∧ g Z ∉ f Z) → f' 0 = f X → (∀m, nat_p m → f' (ordsucc m) = f (f' m)) → (∀m, nat_p m → f' m ⊆ X ∧ infinite (f' m)) ∧ (∀m m' ∈ ω, m ⊆ m' → f' m' ⊆ f' m) ∧ (∀m m' ∈ ω, g (f' m) = g (f' m') → m = m')
In Proofgold the corresponding term root is c8e826... and proposition id is c1cc5f...
Proof:
The rest of this subproof is missing.
∎
Theorem. (infiniteRamsey)
∀c, nat_p c → ∀n, nat_p n → ∀X, infinite X → ∀C : set → set, (∀Y ⊆ X, equip Y n → C Y ∈ c) → ∃H ⊆ X, ∃i ∈ c, infinite H ∧ ∀Y ⊆ H, equip Y n → C Y = i
In Proofgold the corresponding term root is f92f8f... and proposition id is 865fc1...
Proof:
The rest of this subproof is missing.
∎
End of Section InfiniteRamsey
Definition. We define Inj1 to be In_rec_i (λX f ⇒ {0} ∪ {f x|x ∈ X}) of type set → set.
In Proofgold the corresponding term root is 8f0026... and object id is 7736ca...
Theorem. (Inj1_eq)
∀X : set, Inj1 X = {0} ∪ {Inj1 x|x ∈ X}
In Proofgold the corresponding term root is 73d189... and proposition id is b80457...
Proof:
The rest of this subproof is missing.
∎
Theorem. (Inj1I1)
∀X : set, 0 ∈ Inj1 X
In Proofgold the corresponding term root is 3c2cc4... and proposition id is 4ccac3...
Proof:
The rest of this subproof is missing.
∎
Theorem. (Inj1I2)
∀X x : set, x ∈ X → Inj1 x ∈ Inj1 X
In Proofgold the corresponding term root is b6f23b... and proposition id is 086475...
Proof:
The rest of this subproof is missing.
∎
Theorem. (Inj1E)
∀X y : set, y ∈ Inj1 X → y = 0 ∨ ∃x ∈ X, y = Inj1 x
In Proofgold the corresponding term root is d4177b... and proposition id is 259217...
Proof:
The rest of this subproof is missing.
∎
Theorem. (Inj1NE1)
∀x : set, Inj1 x ≠ 0
In Proofgold the corresponding term root is 5e3197... and proposition id is 09bd10...
Proof:
The rest of this subproof is missing.
∎
Theorem. (Inj1NE2)
∀x : set, Inj1 x ∉ {0}
In Proofgold the corresponding term root is 9e94b1... and proposition id is 3c4dd0...
Proof:
The rest of this subproof is missing.
∎
Definition. We define Inj0 to be λX ⇒ {Inj1 x|x ∈ X} of type set → set.
In Proofgold the corresponding term root is 814321... and object id is 7d54be...
Theorem. (Inj0I)
∀X x : set, x ∈ X → Inj1 x ∈ Inj0 X
In Proofgold the corresponding term root is 7fc2e9... and proposition id is 565378...
Proof:
The rest of this subproof is missing.
∎
Theorem. (Inj0E)
∀X y : set, y ∈ Inj0 X → ∃x : set, x ∈ X ∧ y = Inj1 x
In Proofgold the corresponding term root is 416684... and proposition id is 855213...
Proof:
The rest of this subproof is missing.
∎
Definition. We define Unj to be In_rec_i (λX f ⇒ {f x|x ∈ X ∖ {0}}) of type set → set.
In Proofgold the corresponding term root is f202c1... and object id is 2e1675...
Theorem. (Unj_eq)
∀X : set, Unj X = {Unj x|x ∈ X ∖ {0}}
In Proofgold the corresponding term root is 3b1aa6... and proposition id is b4c852...
Proof:
The rest of this subproof is missing.
∎
Theorem. (Unj_Inj1_eq)
∀X : set, Unj (Inj1 X) = X
In Proofgold the corresponding term root is e2cdfc... and proposition id is ae00e1...
Proof:
The rest of this subproof is missing.
∎
Theorem. (Inj1_inj)
∀X Y : set, Inj1 X = Inj1 Y → X = Y
In Proofgold the corresponding term root is ce8b66... and proposition id is 3e918a...
Proof:
The rest of this subproof is missing.
∎
Theorem. (Unj_Inj0_eq)
∀X : set, Unj (Inj0 X) = X
In Proofgold the corresponding term root is f3e39c... and proposition id is 3808ad...
Proof:
The rest of this subproof is missing.
∎
Theorem. (Inj0_inj)
∀X Y : set, Inj0 X = Inj0 Y → X = Y
In Proofgold the corresponding term root is 662f53... and proposition id is 29f945...
Proof:
The rest of this subproof is missing.
∎
Theorem. (Inj0_0)
Inj0 0 = 0
In Proofgold the corresponding term root is 171f47... and proposition id is 57f5af...
Proof:
The rest of this subproof is missing.
∎
Theorem. (Inj0_Inj1_neq)
∀X Y : set, Inj0 X ≠ Inj1 Y
In Proofgold the corresponding term root is 21bb03... and proposition id is 2b14d8...
Proof:
The rest of this subproof is missing.
∎
Definition. We define setsum to be λX Y ⇒ {Inj0 x|x ∈ X} ∪ {Inj1 y|y ∈ Y} of type set → set → set.
In Proofgold the corresponding term root is afe37e... and object id is 5b7e09...
Notation. We use + as an infix operator with priority 450 and which associates to the left corresponding to applying term setsum.
Theorem. (Inj0_setsum)
∀X Y x : set, x ∈ X → Inj0 x ∈ X + Y
In Proofgold the corresponding term root is 1c6408... and proposition id is 246ac5...
Proof:
The rest of this subproof is missing.
∎
Theorem. (Inj1_setsum)
∀X Y y : set, y ∈ Y → Inj1 y ∈ X + Y
In Proofgold the corresponding term root is d47a0f... and proposition id is 2b4f94...
Proof:
The rest of this subproof is missing.
∎
Theorem. (setsum_Inj_inv)
∀X Y z : set, z ∈ X + Y → (∃x ∈ X, z = Inj0 x) ∨ (∃y ∈ Y, z = Inj1 y)
In Proofgold the corresponding term root is 645a6c... and proposition id is a36400...
Proof:
The rest of this subproof is missing.
∎
Theorem. (Inj0_setsum_0L)
∀X : set, 0 + X = Inj0 X
In Proofgold the corresponding term root is 0be1e7... and proposition id is 36fabd...
Proof:
The rest of this subproof is missing.
∎
Theorem. (Inj1_setsum_1L)
∀X : set, 1 + X = Inj1 X
In Proofgold the corresponding term root is f8d721... and proposition id is 45b1c8...
Proof:
The rest of this subproof is missing.
∎
Beginning of Section pair_setsum
Let pair ≝ setsum
Definition. We define proj0 to be λZ ⇒ {Unj z|z ∈ Z, ∃x : set, Inj0 x = z} of type set → set.
In Proofgold the corresponding term root is 21a488... and object id is 9c5e4e...
Definition. We define proj1 to be λZ ⇒ {Unj z|z ∈ Z, ∃y : set, Inj1 y = z} of type set → set.
In Proofgold the corresponding term root is 7aba21... and object id is 7fd81e...
Theorem. (Inj0_pair_0_eq)
Inj0 = pair 0
In Proofgold the corresponding term root is 0fe601... and proposition id is 490c2a...
Proof:
The rest of this subproof is missing.
∎
Theorem. (Inj1_pair_1_eq)
Inj1 = pair 1
In Proofgold the corresponding term root is 0bbde8... and proposition id is 8272d7...
Proof:
The rest of this subproof is missing.
∎
Theorem. (pairI0)
∀X Y x, x ∈ X → pair 0 x ∈ pair X Y
In Proofgold the corresponding term root is 21a414... and proposition id is 34813b...
Proof:
The rest of this subproof is missing.
∎
Theorem. (pairI1)
∀X Y y, y ∈ Y → pair 1 y ∈ pair X Y
In Proofgold the corresponding term root is bdaab5... and proposition id is b5248b...
Proof:
The rest of this subproof is missing.
∎
Theorem. (pairE)
∀X Y z, z ∈ pair X Y → (∃x ∈ X, z = pair 0 x) ∨ (∃y ∈ Y, z = pair 1 y)
In Proofgold the corresponding term root is aac610... and proposition id is 0afdeb...
Proof:
The rest of this subproof is missing.
∎
Theorem. (pairE0)
∀X Y x, pair 0 x ∈ pair X Y → x ∈ X
In Proofgold the corresponding term root is 042f30... and proposition id is f50b25...
Proof:
The rest of this subproof is missing.
∎
Theorem. (pairE1)
∀X Y y, pair 1 y ∈ pair X Y → y ∈ Y
In Proofgold the corresponding term root is 505cfc... and proposition id is 30a181...
Proof:
The rest of this subproof is missing.
∎
Theorem. (proj0I)
∀w u : set, pair 0 u ∈ w → u ∈ proj0 w
In Proofgold the corresponding term root is be55b9... and proposition id is 840c8d...
Proof:
The rest of this subproof is missing.
∎
Theorem. (proj0E)
∀w u : set, u ∈ proj0 w → pair 0 u ∈ w
In Proofgold the corresponding term root is b9afa9... and proposition id is 5b7f58...
Proof:
The rest of this subproof is missing.
∎
Theorem. (proj1I)
∀w u : set, pair 1 u ∈ w → u ∈ proj1 w
In Proofgold the corresponding term root is fbddb7... and proposition id is 595b3c...
Proof:
The rest of this subproof is missing.
∎
Theorem. (proj1E)
∀w u : set, u ∈ proj1 w → pair 1 u ∈ w
In Proofgold the corresponding term root is 06e32e... and proposition id is c3c3e6...
Proof:
The rest of this subproof is missing.
∎
Theorem. (proj0_pair_eq)
∀X Y : set, proj0 (pair X Y) = X
In Proofgold the corresponding term root is d546f3... and proposition id is 0c12fb...
Proof:
The rest of this subproof is missing.
∎
Theorem. (proj1_pair_eq)
∀X Y : set, proj1 (pair X Y) = Y
In Proofgold the corresponding term root is 9f98a8... and proposition id is e0cb0f...
Proof:
The rest of this subproof is missing.
∎
Definition. We define Sigma to be λX Y ⇒ ⋃x ∈ X{pair x y|y ∈ Y x} of type set → (set → set) → set.
In Proofgold the corresponding term root is 8acbb2... and object id is d64bcb...
Notation. We use ∑ x...y [possibly with ascriptions] , B as a binder notation corresponding to a term constructed using Sigma.
Theorem. (Sigma_eta_proj0_proj1)
∀X : set, ∀Y : set → set, ∀z ∈ (∑x ∈ X, Y x), pair (proj0 z) (proj1 z) = z ∧ proj0 z ∈ X ∧ proj1 z ∈ Y (proj0 z)
In Proofgold the corresponding term root is afe962... and proposition id is 77ba8e...
Proof:
The rest of this subproof is missing.
∎
Theorem. (proj0_Sigma)
∀X : set, ∀Y : set → set, ∀z : set, z ∈ (∑x ∈ X, Y x) → proj0 z ∈ X
In Proofgold the corresponding term root is ae12a7... and proposition id is 506869...
Proof:
The rest of this subproof is missing.
∎
Theorem. (proj1_Sigma)
∀X : set, ∀Y : set → set, ∀z : set, z ∈ (∑x ∈ X, Y x) → proj1 z ∈ Y (proj0 z)
In Proofgold the corresponding term root is 553db6... and proposition id is d79917...
Proof:
The rest of this subproof is missing.
∎
Theorem. (pair_Sigma)
∀X : set, ∀Y : set → set, ∀x ∈ X, ∀y ∈ Y x, pair x y ∈ ∑x ∈ X, Y x
In Proofgold the corresponding term root is 0d8645... and proposition id is 1456cb...
Proof:
The rest of this subproof is missing.
∎
Theorem. (pair_Sigma_E1)
∀X : set, ∀Y : set → set, ∀x y : set, pair x y ∈ (∑x ∈ X, Y x) → y ∈ Y x
In Proofgold the corresponding term root is c294a7... and proposition id is 81d8b0...
Proof:
The rest of this subproof is missing.
∎
Theorem. (Sigma_E)
∀X : set, ∀Y : set → set, ∀z : set, z ∈ (∑x ∈ X, Y x) → ∃x ∈ X, ∃y ∈ Y x, z = pair x y
In Proofgold the corresponding term root is 5b1bd2... and proposition id is 8ecfab...
Proof:
The rest of this subproof is missing.
∎
Definition. We define setprod to be λX Y : set ⇒ ∑x ∈ X, Y of type set → set → set.
In Proofgold the corresponding term root is fc0b60... and object id is 107b6b...
Notation. We use ⨯ as an infix operator with priority 440 and which associates to the left corresponding to applying term setprod.
Let lam : set → (set → set) → set ≝ Sigma
Definition. We define ap to be λf x ⇒ {proj1 z|z ∈ f, ∃y : set, z = pair x y} of type set → set → set.
In Proofgold the corresponding term root is c7aaa6... and object id is a4d82c...
Notation. When x is a set, a term x y is notation for ap x y.
Notation. λ x ∈ A ⇒ B is notation for the set Sigma A (λ x : set ⇒ B).
Notation. We now use n-tuple notation (a0,...,an-1) for n ≥ 2 for λ i ∈ n . if i = 0 then a0 else ... if i = n-2 then an-2 else an-1.
Theorem. (lamI)
∀X : set, ∀F : set → set, ∀x ∈ X, ∀y ∈ F x, pair x y ∈ λx ∈ X ⇒ F x
In Proofgold the corresponding term root is 0d8645... and proposition id is 1456cb...
Proof:
The rest of this subproof is missing.
∎
Theorem. (lamE)
∀X : set, ∀F : set → set, ∀z : set, z ∈ (λx ∈ X ⇒ F x) → ∃x ∈ X, ∃y ∈ F x, z = pair x y
In Proofgold the corresponding term root is 5b1bd2... and proposition id is 8ecfab...
Proof:
The rest of this subproof is missing.
∎
Theorem. (apI)
∀f x y, pair x y ∈ f → y ∈ f x
In Proofgold the corresponding term root is 64667c... and proposition id is 770816...
Proof:
The rest of this subproof is missing.
∎
Theorem. (apE)
∀f x y, y ∈ f x → pair x y ∈ f
In Proofgold the corresponding term root is 85693a... and proposition id is c08565...
Proof:
The rest of this subproof is missing.
∎
Theorem. (beta)
∀X : set, ∀F : set → set, ∀x : set, x ∈ X → (λx ∈ X ⇒ F x) x = F x
In Proofgold the corresponding term root is 95991d... and proposition id is b5182d...
Proof:
The rest of this subproof is missing.
∎
Theorem. (proj0_ap_0)
∀u, proj0 u = u 0
In Proofgold the corresponding term root is 588f37... and proposition id is d4f351...
Proof:
The rest of this subproof is missing.
∎
Theorem. (proj1_ap_1)
∀u, proj1 u = u 1
In Proofgold the corresponding term root is d116b3... and proposition id is 2328b0...
Proof:
The rest of this subproof is missing.
∎
Theorem. (pair_ap_0)
∀x y : set, (pair x y) 0 = x
In Proofgold the corresponding term root is ac34b9... and proposition id is bb0477...
Proof:
The rest of this subproof is missing.
∎
Theorem. (pair_ap_1)
∀x y : set, (pair x y) 1 = y
In Proofgold the corresponding term root is 2274fc... and proposition id is cce3d2...
Proof:
The rest of this subproof is missing.
∎
Theorem. (ap0_Sigma)
∀X : set, ∀Y : set → set, ∀z : set, z ∈ (∑x ∈ X, Y x) → (z 0) ∈ X
In Proofgold the corresponding term root is 861a79... and proposition id is 1affcc...
Proof:
The rest of this subproof is missing.
∎
Theorem. (ap1_Sigma)
∀X : set, ∀Y : set → set, ∀z : set, z ∈ (∑x ∈ X, Y x) → (z 1) ∈ (Y (z 0))
In Proofgold the corresponding term root is ad3bfe... and proposition id is 520af5...
Proof:
The rest of this subproof is missing.
∎
Definition. We define pair_p to be λu : set ⇒ pair (u 0) (u 1) = u of type set → prop.
In Proofgold the corresponding term root is ade2bb... and object id is 5c8a35...
Theorem. (pair_p_I)
∀x y, pair_p (pair x y)
In Proofgold the corresponding term root is 06d9a2... and proposition id is 8f2fde...
Proof:
The rest of this subproof is missing.
∎
Theorem. (Subq_2_UPair01)
2 ⊆ {0,1}
In Proofgold the corresponding term root is d90884... and proposition id is 94203e...
Proof:
The rest of this subproof is missing.
∎
Theorem. (tuple_pair)
∀x y : set, pair x y = (x,y)
In Proofgold the corresponding term root is c122f6... and proposition id is 18198e...
Proof:
The rest of this subproof is missing.
∎
Definition. We define Pi to be λX Y ⇒ {f ∈ 𝒫 (∑x ∈ X, ⋃ (Y x))|∀x ∈ X, f x ∈ Y x} of type set → (set → set) → set.
In Proofgold the corresponding term root is 40f1c3... and object id is a7b1b0...
Notation. We use ∏ x...y [possibly with ascriptions] , B as a binder notation corresponding to a term constructed using Pi.
Theorem. (PiI)
∀X : set, ∀Y : set → set, ∀f : set, (∀u ∈ f, pair_p u ∧ u 0 ∈ X) → (∀x ∈ X, f x ∈ Y x) → f ∈ ∏x ∈ X, Y x
In Proofgold the corresponding term root is 2b7ef4... and proposition id is 1f8f6f...
Proof:
The rest of this subproof is missing.
∎
Theorem. (lam_Pi)
∀X : set, ∀Y : set → set, ∀F : set → set, (∀x ∈ X, F x ∈ Y x) → (λx ∈ X ⇒ F x) ∈ (∏x ∈ X, Y x)
In Proofgold the corresponding term root is c0b88f... and proposition id is 3733f8...
Proof:
The rest of this subproof is missing.
∎
Theorem. (ap_Pi)
∀X : set, ∀Y : set → set, ∀f : set, ∀x : set, f ∈ (∏x ∈ X, Y x) → x ∈ X → f x ∈ Y x
In Proofgold the corresponding term root is 7c3617... and proposition id is cb962e...
Proof:
The rest of this subproof is missing.
∎
Definition. We define setexp to be λX Y : set ⇒ ∏y ∈ Y, X of type set → set → set.
In Proofgold the corresponding term root is 1de7fc... and object id is 021ae0...
Notation. We use :^: as an infix operator with priority 430 and which associates to the left corresponding to applying term setexp.
Theorem. (pair_tuple_fun)
pair = (λx y ⇒ (x,y))
In Proofgold the corresponding term root is b02dfa... and proposition id is ed37d0...
Proof:
The rest of this subproof is missing.
∎
Beginning of Section Tuples
Variable x0 x1 : set
Theorem. (tuple_2_0_eq)
(x0,x1) 0 = x0
In Proofgold the corresponding term root is 42d452... and proposition id is ad9037...
Proof:
The rest of this subproof is missing.
∎
Theorem. (tuple_2_1_eq)
(x0,x1) 1 = x1
In Proofgold the corresponding term root is 596569... and proposition id is 2c0ad1...
Proof:
The rest of this subproof is missing.
∎
End of Section Tuples
Theorem. (ReplEq_setprod_ext)
∀X Y, ∀F G : set → set → set, (∀x ∈ X, ∀y ∈ Y, F x y = G x y) → {F (w 0) (w 1)|w ∈ X ⨯ Y} = {G (w 0) (w 1)|w ∈ X ⨯ Y}
In Proofgold the corresponding term root is c486ab... and proposition id is f689a0...
Proof:
The rest of this subproof is missing.
∎
Theorem. (lamI2)
∀X, ∀F : set → set, ∀x ∈ X, ∀y ∈ F x, (x,y) ∈ λx ∈ X ⇒ F x
In Proofgold the corresponding term root is d4a9d9... and proposition id is de381f...
Proof:
The rest of this subproof is missing.
∎
Theorem. (tuple_2_Sigma)
∀X : set, ∀Y : set → set, ∀x ∈ X, ∀y ∈ Y x, (x,y) ∈ ∑x ∈ X, Y x
In Proofgold the corresponding term root is d4a9d9... and proposition id is de381f...
Proof:
The rest of this subproof is missing.
∎
Theorem. (tuple_2_setprod)
∀X : set, ∀Y : set, ∀x ∈ X, ∀y ∈ Y, (x,y) ∈ X ⨯ Y
In Proofgold the corresponding term root is e907b8... and proposition id is 1f9d7f...
Proof:
The rest of this subproof is missing.
∎
End of Section pair_setsum
Notation. We use ∑ x...y [possibly with ascriptions] , B as a binder notation corresponding to a term constructed using Sigma.
Notation. We use ⨯ as an infix operator with priority 440 and which associates to the left corresponding to applying term setprod.
Notation. We use ∏ x...y [possibly with ascriptions] , B as a binder notation corresponding to a term constructed using Pi.
Notation. We use :^: as an infix operator with priority 430 and which associates to the left corresponding to applying term setexp.
Definition. We define DescrR_i_io_1 to be λR ⇒ Eps_i (λx ⇒ (∃y : set → prop, R x y) ∧ (∀y z : set → prop, R x y → R x z → y = z)) of type (set → (set → prop) → prop) → set.
In Proofgold the corresponding term root is 1d3fd4... and object id is 82aecc...
Definition. We define DescrR_i_io_2 to be λR ⇒ Descr_Vo1 (λy ⇒ R (DescrR_i_io_1 R) y) of type (set → (set → prop) → prop) → set → prop.
In Proofgold the corresponding term root is 768eb2... and object id is 3f82bc...
Theorem. (DescrR_i_io_12)
∀R : set → (set → prop) → prop, (∃x, (∃y : set → prop, R x y) ∧ (∀y z : set → prop, R x y → R x z → y = z)) → R (DescrR_i_io_1 R) (DescrR_i_io_2 R)
In Proofgold the corresponding term root is a4fae2... and proposition id is da4f02...
Proof:
The rest of this subproof is missing.
∎
Definition. We define PNoEq_ to be λalpha p q ⇒ ∀beta ∈ alpha, p beta ↔ q beta of type set → (set → prop) → (set → prop) → prop.
In Proofgold the corresponding term root is d7d959... and object id is 144b65...
Theorem. (PNoEq_ref_)
∀alpha, ∀p : set → prop, PNoEq_ alpha p p
In Proofgold the corresponding term root is c27ebb... and proposition id is df4cf6...
Proof:
The rest of this subproof is missing.
∎
Theorem. (PNoEq_sym_)
∀alpha, ∀p q : set → prop, PNoEq_ alpha p q → PNoEq_ alpha q p
In Proofgold the corresponding term root is b813f3... and proposition id is bfda0b...
Proof:
The rest of this subproof is missing.
∎
Theorem. (PNoEq_tra_)
∀alpha, ∀p q r : set → prop, PNoEq_ alpha p q → PNoEq_ alpha q r → PNoEq_ alpha p r
In Proofgold the corresponding term root is dded4c... and proposition id is fc92f3...
Proof:
The rest of this subproof is missing.
∎
Theorem. (PNoEq_antimon_)
∀p q : set → prop, ∀alpha, ordinal alpha → ∀beta ∈ alpha, PNoEq_ alpha p q → PNoEq_ beta p q
In Proofgold the corresponding term root is 8868e6... and proposition id is 094ae2...
Proof:
The rest of this subproof is missing.
∎
Definition. We define PNoLt_ to be λalpha p q ⇒ ∃beta ∈ alpha, PNoEq_ beta p q ∧ ¬ p beta ∧ q beta of type set → (set → prop) → (set → prop) → prop.
In Proofgold the corresponding term root is 34de68... and object id is c95964...
Theorem. (PNoLt_E_)
∀alpha, ∀p q : set → prop, PNoLt_ alpha p q → ∀R : prop, (∀beta, beta ∈ alpha → PNoEq_ beta p q → ¬ p beta → q beta → R) → R
In Proofgold the corresponding term root is 9ba543... and proposition id is 45aec8...
Proof:
The rest of this subproof is missing.
∎
Theorem. (PNoLt_irref_)
∀alpha, ∀p : set → prop, ¬ PNoLt_ alpha p p
In Proofgold the corresponding term root is fcb502... and proposition id is 987dff...
Proof:
The rest of this subproof is missing.
∎
Theorem. (PNoLt_mon_)
∀p q : set → prop, ∀alpha, ordinal alpha → ∀beta ∈ alpha, PNoLt_ beta p q → PNoLt_ alpha p q
In Proofgold the corresponding term root is b1c3cf... and proposition id is d8d0f0...
Proof:
The rest of this subproof is missing.
∎
Theorem. (PNoLt_trichotomy_or_)
∀p q : set → prop, ∀alpha, ordinal alpha → PNoLt_ alpha p q ∨ PNoEq_ alpha p q ∨ PNoLt_ alpha q p
In Proofgold the corresponding term root is 664408... and proposition id is 8d3824...
Proof:
The rest of this subproof is missing.
∎
Definition. We define PNoLt to be λalpha p beta q ⇒ PNoLt_ (alpha ∩ beta) p q ∨ alpha ∈ beta ∧ PNoEq_ alpha p q ∧ q alpha ∨ beta ∈ alpha ∧ PNoEq_ beta p q ∧ ¬ p beta of type set → (set → prop) → set → (set → prop) → prop.
In Proofgold the corresponding term root is 8f57a0... and object id is f2094c...
Theorem. (PNoLtI1)
∀alpha beta, ∀p q : set → prop, PNoLt_ (alpha ∩ beta) p q → PNoLt alpha p beta q
In Proofgold the corresponding term root is c5735c... and proposition id is ecbc71...
Proof:
The rest of this subproof is missing.
∎
Theorem. (PNoLtI2)
∀alpha beta, ∀p q : set → prop, alpha ∈ beta → PNoEq_ alpha p q → q alpha → PNoLt alpha p beta q
In Proofgold the corresponding term root is 419b53... and proposition id is 030093...
Proof:
The rest of this subproof is missing.
∎
Theorem. (PNoLtI3)
∀alpha beta, ∀p q : set → prop, beta ∈ alpha → PNoEq_ beta p q → ¬ p beta → PNoLt alpha p beta q
In Proofgold the corresponding term root is c392cb... and proposition id is 410442...
Proof:
The rest of this subproof is missing.
∎
Theorem. (PNoLtE)
∀alpha beta, ∀p q : set → prop, PNoLt alpha p beta q → ∀R : prop, (PNoLt_ (alpha ∩ beta) p q → R) → (alpha ∈ beta → PNoEq_ alpha p q → q alpha → R) → (beta ∈ alpha → PNoEq_ beta p q → ¬ p beta → R) → R
In Proofgold the corresponding term root is f7e6e3... and proposition id is af36ce...
Proof:
The rest of this subproof is missing.
∎
Theorem. (PNoLt_irref)
∀alpha, ∀p : set → prop, ¬ PNoLt alpha p alpha p
In Proofgold the corresponding term root is ff1a80... and proposition id is d4c379...
Proof:
The rest of this subproof is missing.
∎
Theorem. (PNoLt_trichotomy_or)
∀alpha beta, ∀p q : set → prop, ordinal alpha → ordinal beta → PNoLt alpha p beta q ∨ alpha = beta ∧ PNoEq_ alpha p q ∨ PNoLt beta q alpha p
In Proofgold the corresponding term root is 4f39e4... and proposition id is 90b387...
Proof:
The rest of this subproof is missing.
∎
Theorem. (PNoLtEq_tra)
∀alpha beta, ordinal alpha → ordinal beta → ∀p q r : set → prop, PNoLt alpha p beta q → PNoEq_ beta q r → PNoLt alpha p beta r
In Proofgold the corresponding term root is 2e082a... and proposition id is 2f2a82...
Proof:
The rest of this subproof is missing.
∎
Theorem. (PNoEqLt_tra)
∀alpha beta, ordinal alpha → ordinal beta → ∀p q r : set → prop, PNoEq_ alpha p q → PNoLt alpha q beta r → PNoLt alpha p beta r
In Proofgold the corresponding term root is fe144b... and proposition id is f07654...
Proof:
The rest of this subproof is missing.
∎
Theorem. (PNoLt_tra)
∀alpha beta gamma, ordinal alpha → ordinal beta → ordinal gamma → ∀p q r : set → prop, PNoLt alpha p beta q → PNoLt beta q gamma r → PNoLt alpha p gamma r
In Proofgold the corresponding term root is b72127... and proposition id is 8eff85...
Proof:
The rest of this subproof is missing.
∎
Definition. We define PNoLe to be λalpha p beta q ⇒ PNoLt alpha p beta q ∨ alpha = beta ∧ PNoEq_ alpha p q of type set → (set → prop) → set → (set → prop) → prop.
In Proofgold the corresponding term root is ac6311... and object id is 770cf7...
Theorem. (PNoLeI1)
∀alpha beta, ∀p q : set → prop, PNoLt alpha p beta q → PNoLe alpha p beta q
In Proofgold the corresponding term root is 1f0528... and proposition id is 674a2c...
Proof:
The rest of this subproof is missing.
∎
Theorem. (PNoLeI2)
∀alpha, ∀p q : set → prop, PNoEq_ alpha p q → PNoLe alpha p alpha q
In Proofgold the corresponding term root is b45332... and proposition id is 0b2157...
Proof:
The rest of this subproof is missing.
∎
Theorem. (PNoLe_ref)
∀alpha, ∀p : set → prop, PNoLe alpha p alpha p
In Proofgold the corresponding term root is 1b91bd... and proposition id is 613386...
Proof:
The rest of this subproof is missing.
∎
Theorem. (PNoLe_antisym)
∀alpha beta, ordinal alpha → ordinal beta → ∀p q : set → prop, PNoLe alpha p beta q → PNoLe beta q alpha p → alpha = beta ∧ PNoEq_ alpha p q
In Proofgold the corresponding term root is 934d7a... and proposition id is 4e3594...
Proof:
The rest of this subproof is missing.
∎
Theorem. (PNoLtLe_tra)
∀alpha beta gamma, ordinal alpha → ordinal beta → ordinal gamma → ∀p q r : set → prop, PNoLt alpha p beta q → PNoLe beta q gamma r → PNoLt alpha p gamma r
In Proofgold the corresponding term root is 156586... and proposition id is 09246b...
Proof:
The rest of this subproof is missing.
∎
Theorem. (PNoLeLt_tra)
∀alpha beta gamma, ordinal alpha → ordinal beta → ordinal gamma → ∀p q r : set → prop, PNoLe alpha p beta q → PNoLt beta q gamma r → PNoLt alpha p gamma r
In Proofgold the corresponding term root is a260ca... and proposition id is ad6971...
Proof:
The rest of this subproof is missing.
∎
Theorem. (PNoEqLe_tra)
∀alpha beta, ordinal alpha → ordinal beta → ∀p q r : set → prop, PNoEq_ alpha p q → PNoLe alpha q beta r → PNoLe alpha p beta r
In Proofgold the corresponding term root is 7d3ff8... and proposition id is 002ebf...
Proof:
The rest of this subproof is missing.
∎
Theorem. (PNoLe_tra)
∀alpha beta gamma, ordinal alpha → ordinal beta → ordinal gamma → ∀p q r : set → prop, PNoLe alpha p beta q → PNoLe beta q gamma r → PNoLe alpha p gamma r
In Proofgold the corresponding term root is 70b92b... and proposition id is 4b557d...
Proof:
The rest of this subproof is missing.
∎
Definition. We define PNo_downc to be λL alpha p ⇒ ∃beta, ordinal beta ∧ ∃q : set → prop, L beta q ∧ PNoLe alpha p beta q of type (set → (set → prop) → prop) → set → (set → prop) → prop.
In Proofgold the corresponding term root is 93dab1... and object id is f6c962...
Definition. We define PNo_upc to be λR alpha p ⇒ ∃beta, ordinal beta ∧ ∃q : set → prop, R beta q ∧ PNoLe beta q alpha p of type (set → (set → prop) → prop) → set → (set → prop) → prop.
In Proofgold the corresponding term root is 543712... and object id is b97935...
Theorem. (PNoLe_downc)
∀L : set → (set → prop) → prop, ∀alpha beta, ∀p q : set → prop, ordinal alpha → ordinal beta → PNo_downc L alpha p → PNoLe beta q alpha p → PNo_downc L beta q
In Proofgold the corresponding term root is 6b1f0d... and proposition id is cd130e...
Proof:
The rest of this subproof is missing.
∎
Theorem. (PNo_downc_ref)
∀L : set → (set → prop) → prop, ∀alpha, ordinal alpha → ∀p : set → prop, L alpha p → PNo_downc L alpha p
In Proofgold the corresponding term root is 347aa9... and proposition id is 69dfcb...
Proof:
The rest of this subproof is missing.
∎
Theorem. (PNo_upc_ref)
∀R : set → (set → prop) → prop, ∀alpha, ordinal alpha → ∀p : set → prop, R alpha p → PNo_upc R alpha p
In Proofgold the corresponding term root is 403942... and proposition id is c4bf71...
Proof:
The rest of this subproof is missing.
∎
Theorem. (PNoLe_upc)
∀R : set → (set → prop) → prop, ∀alpha beta, ∀p q : set → prop, ordinal alpha → ordinal beta → PNo_upc R alpha p → PNoLe alpha p beta q → PNo_upc R beta q
In Proofgold the corresponding term root is 111b74... and proposition id is eb9ed5...
Proof:
The rest of this subproof is missing.
∎
Definition. We define PNoLt_pwise to be λL R ⇒ ∀gamma, ordinal gamma → ∀p : set → prop, L gamma p → ∀delta, ordinal delta → ∀q : set → prop, R delta q → PNoLt gamma p delta q of type (set → (set → prop) → prop) → (set → (set → prop) → prop) → prop.
In Proofgold the corresponding term root is ae448b... and object id is 123808...
Theorem. (PNoLt_pwise_downc_upc)
∀L R : set → (set → prop) → prop, PNoLt_pwise L R → PNoLt_pwise (PNo_downc L) (PNo_upc R)
In Proofgold the corresponding term root is 2b8e3b... and proposition id is 2f244c...
Proof:
The rest of this subproof is missing.
∎
Definition. We define PNo_rel_strict_upperbd to be λL alpha p ⇒ ∀beta ∈ alpha, ∀q : set → prop, PNo_downc L beta q → PNoLt beta q alpha p of type (set → (set → prop) → prop) → set → (set → prop) → prop.
In Proofgold the corresponding term root is dc1665... and object id is 3e1663...
Definition. We define PNo_rel_strict_lowerbd to be λR alpha p ⇒ ∀beta ∈ alpha, ∀q : set → prop, PNo_upc R beta q → PNoLt alpha p beta q of type (set → (set → prop) → prop) → set → (set → prop) → prop.
In Proofgold the corresponding term root is e5a4b6... and object id is ff3150...
Definition. We define PNo_rel_strict_imv to be λL R alpha p ⇒ PNo_rel_strict_upperbd L alpha p ∧ PNo_rel_strict_lowerbd R alpha p of type (set → (set → prop) → prop) → (set → (set → prop) → prop) → set → (set → prop) → prop.
In Proofgold the corresponding term root is 64ce99... and object id is 5b35fb...
Theorem. (PNoEq_rel_strict_upperbd)
∀L : set → (set → prop) → prop, ∀alpha, ordinal alpha → ∀p q : set → prop, PNoEq_ alpha p q → PNo_rel_strict_upperbd L alpha p → PNo_rel_strict_upperbd L alpha q
In Proofgold the corresponding term root is bb65cf... and proposition id is db4b82...
Proof:
The rest of this subproof is missing.
∎
Theorem. (PNo_rel_strict_upperbd_antimon)
∀L : set → (set → prop) → prop, ∀alpha, ordinal alpha → ∀p : set → prop, ∀beta ∈ alpha, PNo_rel_strict_upperbd L alpha p → PNo_rel_strict_upperbd L beta p
In Proofgold the corresponding term root is 9986af... and proposition id is ed53da...
Proof:
The rest of this subproof is missing.
∎
Theorem. (PNoEq_rel_strict_lowerbd)
∀R : set → (set → prop) → prop, ∀alpha, ordinal alpha → ∀p q : set → prop, PNoEq_ alpha p q → PNo_rel_strict_lowerbd R alpha p → PNo_rel_strict_lowerbd R alpha q
In Proofgold the corresponding term root is 76080e... and proposition id is 614e58...
Proof:
The rest of this subproof is missing.
∎
Theorem. (PNo_rel_strict_lowerbd_antimon)
∀R : set → (set → prop) → prop, ∀alpha, ordinal alpha → ∀p : set → prop, ∀beta ∈ alpha, PNo_rel_strict_lowerbd R alpha p → PNo_rel_strict_lowerbd R beta p
In Proofgold the corresponding term root is 27f253... and proposition id is 344b63...
Proof:
The rest of this subproof is missing.
∎
Theorem. (PNoEq_rel_strict_imv)
∀L R : set → (set → prop) → prop, ∀alpha, ordinal alpha → ∀p q : set → prop, PNoEq_ alpha p q → PNo_rel_strict_imv L R alpha p → PNo_rel_strict_imv L R alpha q
In Proofgold the corresponding term root is e3f251... and proposition id is 7ae13d...
Proof:
The rest of this subproof is missing.
∎
Theorem. (PNo_rel_strict_imv_antimon)
∀L R : set → (set → prop) → prop, ∀alpha, ordinal alpha → ∀p : set → prop, ∀beta ∈ alpha, PNo_rel_strict_imv L R alpha p → PNo_rel_strict_imv L R beta p
In Proofgold the corresponding term root is 2434dc... and proposition id is 87e0ce...
Proof:
The rest of this subproof is missing.
∎
Definition. We define PNo_rel_strict_uniq_imv to be λL R alpha p ⇒ PNo_rel_strict_imv L R alpha p ∧ ∀q : set → prop, PNo_rel_strict_imv L R alpha q → PNoEq_ alpha p q of type (set → (set → prop) → prop) → (set → (set → prop) → prop) → set → (set → prop) → prop.
In Proofgold the corresponding term root is 23c9d9... and object id is 56ed1b...
Definition. We define PNo_rel_strict_split_imv to be λL R alpha p ⇒ PNo_rel_strict_imv L R (ordsucc alpha) (λdelta ⇒ p delta ∧ delta ≠ alpha) ∧ PNo_rel_strict_imv L R (ordsucc alpha) (λdelta ⇒ p delta ∨ delta = alpha) of type (set → (set → prop) → prop) → (set → (set → prop) → prop) → set → (set → prop) → prop.
In Proofgold the corresponding term root is 217d17... and object id is dd9cee...
Theorem. (PNo_extend0_eq)
∀alpha, ∀p : set → prop, PNoEq_ alpha p (λdelta ⇒ p delta ∧ delta ≠ alpha)
In Proofgold the corresponding term root is 2d441e... and proposition id is 5ed08e...
Proof:
The rest of this subproof is missing.
∎
Theorem. (PNo_extend1_eq)
∀alpha, ∀p : set → prop, PNoEq_ alpha p (λdelta ⇒ p delta ∨ delta = alpha)
In Proofgold the corresponding term root is 8eb99c... and proposition id is 4cac1a...
Proof:
The rest of this subproof is missing.
∎
Theorem. (PNo_rel_imv_ex)
∀L R : set → (set → prop) → prop, PNoLt_pwise L R → ∀alpha, ordinal alpha → (∃p : set → prop, PNo_rel_strict_uniq_imv L R alpha p) ∨ (∃tau ∈ alpha, ∃p : set → prop, PNo_rel_strict_split_imv L R tau p)
In Proofgold the corresponding term root is 3e3224... and proposition id is 749b63...
Proof:
The rest of this subproof is missing.
∎
Definition. We define PNo_lenbdd to be λalpha L ⇒ ∀beta, ∀p : set → prop, L beta p → beta ∈ alpha of type set → (set → (set → prop) → prop) → prop.
In Proofgold the corresponding term root is 009fa7... and object id is a452a6...
Theorem. (PNo_lenbdd_strict_imv_extend0)
∀L R : set → (set → prop) → prop, ∀alpha, ordinal alpha → PNo_lenbdd alpha L → PNo_lenbdd alpha R → ∀p : set → prop, PNo_rel_strict_imv L R alpha p → PNo_rel_strict_imv L R (ordsucc alpha) (λdelta ⇒ p delta ∧ delta ≠ alpha)
In Proofgold the corresponding term root is 17a114... and proposition id is 69327e...
Proof:
The rest of this subproof is missing.
∎
Theorem. (PNo_lenbdd_strict_imv_extend1)
∀L R : set → (set → prop) → prop, ∀alpha, ordinal alpha → PNo_lenbdd alpha L → PNo_lenbdd alpha R → ∀p : set → prop, PNo_rel_strict_imv L R alpha p → PNo_rel_strict_imv L R (ordsucc alpha) (λdelta ⇒ p delta ∨ delta = alpha)
In Proofgold the corresponding term root is ad69ec... and proposition id is 5ff993...
Proof:
The rest of this subproof is missing.
∎
Theorem. (PNo_lenbdd_strict_imv_split)
∀L R : set → (set → prop) → prop, ∀alpha, ordinal alpha → PNo_lenbdd alpha L → PNo_lenbdd alpha R → ∀p : set → prop, PNo_rel_strict_imv L R alpha p → PNo_rel_strict_split_imv L R alpha p
In Proofgold the corresponding term root is 76ae4f... and proposition id is f98998...
Proof:
The rest of this subproof is missing.
∎
Theorem. (PNo_rel_imv_bdd_ex)
∀L R : set → (set → prop) → prop, PNoLt_pwise L R → ∀alpha, ordinal alpha → PNo_lenbdd alpha L → PNo_lenbdd alpha R → ∃beta ∈ ordsucc alpha, ∃p : set → prop, PNo_rel_strict_split_imv L R beta p
In Proofgold the corresponding term root is 9a98e4... and proposition id is 25bcd2...
Proof:
The rest of this subproof is missing.
∎
Definition. We define PNo_strict_upperbd to be λL alpha p ⇒ ∀beta, ordinal beta → ∀q : set → prop, L beta q → PNoLt beta q alpha p of type (set → (set → prop) → prop) → set → (set → prop) → prop.
In Proofgold the corresponding term root is 6913bd... and object id is 3723ab...
Definition. We define PNo_strict_lowerbd to be λR alpha p ⇒ ∀beta, ordinal beta → ∀q : set → prop, R beta q → PNoLt alpha p beta q of type (set → (set → prop) → prop) → set → (set → prop) → prop.
In Proofgold the corresponding term root is 87fac4... and object id is 3606d4...
Definition. We define PNo_strict_imv to be λL R alpha p ⇒ PNo_strict_upperbd L alpha p ∧ PNo_strict_lowerbd R alpha p of type (set → (set → prop) → prop) → (set → (set → prop) → prop) → set → (set → prop) → prop.
In Proofgold the corresponding term root is ee97f6... and object id is bc6c55...
Theorem. (PNoEq_strict_upperbd)
∀L : set → (set → prop) → prop, ∀alpha, ordinal alpha → ∀p q : set → prop, PNoEq_ alpha p q → PNo_strict_upperbd L alpha p → PNo_strict_upperbd L alpha q
In Proofgold the corresponding term root is 9b7b8e... and proposition id is f28b04...
Proof:
The rest of this subproof is missing.
∎
Theorem. (PNoEq_strict_lowerbd)
∀R : set → (set → prop) → prop, ∀alpha, ordinal alpha → ∀p q : set → prop, PNoEq_ alpha p q → PNo_strict_lowerbd R alpha p → PNo_strict_lowerbd R alpha q
In Proofgold the corresponding term root is 4db3f9... and proposition id is 67fa73...
Proof:
The rest of this subproof is missing.
∎
Theorem. (PNoEq_strict_imv)
∀L R : set → (set → prop) → prop, ∀alpha, ordinal alpha → ∀p q : set → prop, PNoEq_ alpha p q → PNo_strict_imv L R alpha p → PNo_strict_imv L R alpha q
In Proofgold the corresponding term root is 1ec352... and proposition id is 5f582f...
Proof:
The rest of this subproof is missing.
∎
Theorem. (PNo_strict_upperbd_imp_rel_strict_upperbd)
∀L : set → (set → prop) → prop, ∀alpha, ordinal alpha → ∀beta ∈ ordsucc alpha, ∀p : set → prop, PNo_strict_upperbd L alpha p → PNo_rel_strict_upperbd L beta p
In Proofgold the corresponding term root is 2898f0... and proposition id is 3cdd3b...
Proof:
The rest of this subproof is missing.
∎
Theorem. (PNo_strict_lowerbd_imp_rel_strict_lowerbd)
∀R : set → (set → prop) → prop, ∀alpha, ordinal alpha → ∀beta ∈ ordsucc alpha, ∀p : set → prop, PNo_strict_lowerbd R alpha p → PNo_rel_strict_lowerbd R beta p
In Proofgold the corresponding term root is 442af5... and proposition id is 04acc2...
Proof:
The rest of this subproof is missing.
∎
Theorem. (PNo_strict_imv_imp_rel_strict_imv)
∀L R : set → (set → prop) → prop, ∀alpha, ordinal alpha → ∀beta ∈ ordsucc alpha, ∀p : set → prop, PNo_strict_imv L R alpha p → PNo_rel_strict_imv L R beta p
In Proofgold the corresponding term root is f4fc71... and proposition id is 46a6aa...
Proof:
The rest of this subproof is missing.
∎
Theorem. (PNo_rel_split_imv_imp_strict_imv)
∀L R : set → (set → prop) → prop, ∀alpha, ordinal alpha → ∀p : set → prop, PNo_rel_strict_split_imv L R alpha p → PNo_strict_imv L R alpha p
In Proofgold the corresponding term root is 72be36... and proposition id is adfdb0...
Proof:
The rest of this subproof is missing.
∎
Theorem. (PNo_lenbdd_strict_imv_ex)
∀L R : set → (set → prop) → prop, PNoLt_pwise L R → ∀alpha, ordinal alpha → PNo_lenbdd alpha L → PNo_lenbdd alpha R → ∃beta ∈ ordsucc alpha, ∃p : set → prop, PNo_strict_imv L R beta p
In Proofgold the corresponding term root is 9485d6... and proposition id is a864a5...
Proof:
The rest of this subproof is missing.
∎
Definition. We define PNo_least_rep to be λL R beta p ⇒ ordinal beta ∧ PNo_strict_imv L R beta p ∧ ∀gamma ∈ beta, ∀q : set → prop, ¬ PNo_strict_imv L R gamma q of type (set → (set → prop) → prop) → (set → (set → prop) → prop) → set → (set → prop) → prop.
In Proofgold the corresponding term root is 0711e2... and object id is e4c707...
Definition. We define PNo_least_rep2 to be λL R beta p ⇒ PNo_least_rep L R beta p ∧ ∀x, x ∉ beta → ¬ p x of type (set → (set → prop) → prop) → (set → (set → prop) → prop) → set → (set → prop) → prop.
In Proofgold the corresponding term root is 1f6dc5... and object id is 09d320...
Theorem. (PNo_strict_imv_pred_eq)
∀L R : set → (set → prop) → prop, PNoLt_pwise L R → ∀alpha, ordinal alpha → ∀p q : set → prop, PNo_least_rep L R alpha p → PNo_strict_imv L R alpha q → ∀beta ∈ alpha, p beta ↔ q beta
In Proofgold the corresponding term root is 5e7822... and proposition id is 72e14d...
Proof:
The rest of this subproof is missing.
∎
Theorem. (PNo_lenbdd_least_rep2_exuniq2)
∀L R : set → (set → prop) → prop, PNoLt_pwise L R → ∀alpha, ordinal alpha → PNo_lenbdd alpha L → PNo_lenbdd alpha R → ∃beta, (∃p : set → prop, PNo_least_rep2 L R beta p) ∧ (∀p q : set → prop, PNo_least_rep2 L R beta p → PNo_least_rep2 L R beta q → p = q)
In Proofgold the corresponding term root is 5060a9... and proposition id is 8b62c5...
Proof:
The rest of this subproof is missing.
∎
Definition. We define PNo_bd to be λL R ⇒ DescrR_i_io_1 (PNo_least_rep2 L R) of type (set → (set → prop) → prop) → (set → (set → prop) → prop) → set.
In Proofgold the corresponding term root is ed76e7... and object id is aa75ef...
Definition. We define PNo_pred to be λL R ⇒ DescrR_i_io_2 (PNo_least_rep2 L R) of type (set → (set → prop) → prop) → (set → (set → prop) → prop) → set → prop.
In Proofgold the corresponding term root is b2d51d... and object id is 2efd18...
Theorem. (PNo_bd_pred_lem)
∀L R : set → (set → prop) → prop, PNoLt_pwise L R → ∀alpha, ordinal alpha → PNo_lenbdd alpha L → PNo_lenbdd alpha R → PNo_least_rep2 L R (PNo_bd L R) (PNo_pred L R)
In Proofgold the corresponding term root is 749b8f... and proposition id is 5de4a3...
Proof:
The rest of this subproof is missing.
∎
Theorem. (PNo_bd_pred)
∀L R : set → (set → prop) → prop, PNoLt_pwise L R → ∀alpha, ordinal alpha → PNo_lenbdd alpha L → PNo_lenbdd alpha R → PNo_least_rep L R (PNo_bd L R) (PNo_pred L R)
In Proofgold the corresponding term root is 42224b... and proposition id is 3309f6...
Proof:
The rest of this subproof is missing.
∎
Theorem. (PNo_bd_In)
∀L R : set → (set → prop) → prop, PNoLt_pwise L R → ∀alpha, ordinal alpha → PNo_lenbdd alpha L → PNo_lenbdd alpha R → PNo_bd L R ∈ ordsucc alpha
In Proofgold the corresponding term root is ad857b... and proposition id is 4719e1...
Proof:
The rest of this subproof is missing.
∎
Beginning of Section TaggedSets
Let tag : set → set ≝ λalpha ⇒ SetAdjoin alpha {1}
Notation. We use ' as a postfix operator with priority 100 corresponding to applying term tag.
Theorem. (not_TransSet_Sing1)
¬ TransSet {1}
In Proofgold the corresponding term root is 8f2d67... and proposition id is be9a70...
Proof:
The rest of this subproof is missing.
∎
Theorem. (not_ordinal_Sing1)
¬ ordinal {1}
In Proofgold the corresponding term root is 99a3f6... and proposition id is db7977...
Proof:
The rest of this subproof is missing.
∎
Theorem. (tagged_not_ordinal)
∀y, ¬ ordinal (y ')
In Proofgold the corresponding term root is f9488c... and proposition id is aaf6e4...
Proof:
The rest of this subproof is missing.
∎
Theorem. (tagged_notin_ordinal)
∀alpha y, ordinal alpha → (y ') ∉ alpha
In Proofgold the corresponding term root is b603ba... and proposition id is c0e2c3...
Proof:
The rest of this subproof is missing.
∎
Theorem. (tagged_eqE_Subq)
∀alpha beta, ordinal alpha → alpha ' = beta ' → alpha ⊆ beta
In Proofgold the corresponding term root is cf032f... and proposition id is ef37f5...
Proof:
The rest of this subproof is missing.
∎
Theorem. (tagged_eqE_eq)
∀alpha beta, ordinal alpha → ordinal beta → alpha ' = beta ' → alpha = beta
In Proofgold the corresponding term root is 1f7dc9... and proposition id is ec1608...
Proof:
The rest of this subproof is missing.
∎
Theorem. (tagged_ReplE)
∀alpha beta, ordinal alpha → ordinal beta → beta ' ∈ {gamma '|gamma ∈ alpha} → beta ∈ alpha
In Proofgold the corresponding term root is a1d43c... and proposition id is 70c250...
Proof:
The rest of this subproof is missing.
∎
Theorem. (ordinal_notin_tagged_Repl)
∀alpha Y, ordinal alpha → alpha ∉ {y '|y ∈ Y}
In Proofgold the corresponding term root is 7f35b5... and proposition id is e7ff3b...
Proof:
The rest of this subproof is missing.
∎
Definition. We define SNoElts_ to be λalpha ⇒ alpha ∪ {beta '|beta ∈ alpha} of type set → set.
In Proofgold the corresponding term root is c0ec73... and object id is 6f0c36...
Theorem. (SNoElts_mon)
∀alpha beta, alpha ⊆ beta → SNoElts_ alpha ⊆ SNoElts_ beta
In Proofgold the corresponding term root is 49a08e... and proposition id is 859611...
Proof:
The rest of this subproof is missing.
∎
Definition. We define SNo_ to be λalpha x ⇒ x ⊆ SNoElts_ alpha ∧ ∀beta ∈ alpha, exactly1of2 (beta ' ∈ x) (beta ∈ x) of type set → set → prop.
In Proofgold the corresponding term root is 4ab7e4... and object id is c16a73...
Definition. We define PSNo to be λalpha p ⇒ {beta ∈ alpha|p beta} ∪ {beta '|beta ∈ alpha, ¬ p beta} of type set → (set → prop) → set.
In Proofgold the corresponding term root is 3657ae... and object id is 0a8351...
Theorem. (PNoEq_PSNo)
∀alpha, ordinal alpha → ∀p : set → prop, PNoEq_ alpha (λbeta ⇒ beta ∈ PSNo alpha p) p
In Proofgold the corresponding term root is c12e02... and proposition id is 688518...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNo_PSNo)
∀alpha, ordinal alpha → ∀p : set → prop, SNo_ alpha (PSNo alpha p)
In Proofgold the corresponding term root is 96e4e2... and proposition id is f03db0...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNo_PSNo_eta_)
∀alpha x, ordinal alpha → SNo_ alpha x → x = PSNo alpha (λbeta ⇒ beta ∈ x)
In Proofgold the corresponding term root is 1642af... and proposition id is 630119...
Proof:
The rest of this subproof is missing.
∎
Definition. We define SNo to be λx ⇒ ∃alpha, ordinal alpha ∧ SNo_ alpha x of type set → prop.
In Proofgold the corresponding term root is 11faa7... and object id is 116b76...
Theorem. (SNo_SNo)
∀alpha, ordinal alpha → ∀z, SNo_ alpha z → SNo z
In Proofgold the corresponding term root is 929b7a... and proposition id is a59341...
Proof:
The rest of this subproof is missing.
∎
Definition. We define SNoLev to be λx ⇒ Eps_i (λalpha ⇒ ordinal alpha ∧ SNo_ alpha x) of type set → set.
In Proofgold the corresponding term root is 293b77... and object id is 83dce3...
Theorem. (SNoLev_uniq_Subq)
∀x alpha beta, ordinal alpha → ordinal beta → SNo_ alpha x → SNo_ beta x → alpha ⊆ beta
In Proofgold the corresponding term root is 2ca114... and proposition id is 65843d...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNoLev_uniq)
∀x alpha beta, ordinal alpha → ordinal beta → SNo_ alpha x → SNo_ beta x → alpha = beta
In Proofgold the corresponding term root is 68f71d... and proposition id is 5d96d4...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNoLev_prop)
∀x, SNo x → ordinal (SNoLev x) ∧ SNo_ (SNoLev x) x
In Proofgold the corresponding term root is a8afcc... and proposition id is 21761c...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNoLev_ordinal)
∀x, SNo x → ordinal (SNoLev x)
In Proofgold the corresponding term root is 7cacc5... and proposition id is f0e9f7...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNoLev_)
∀x, SNo x → SNo_ (SNoLev x) x
In Proofgold the corresponding term root is 4a1033... and proposition id is 49cd5a...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNo_PSNo_eta)
∀x, SNo x → x = PSNo (SNoLev x) (λbeta ⇒ beta ∈ x)
In Proofgold the corresponding term root is 6ffefe... and proposition id is 8c7870...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNoLev_PSNo)
∀alpha, ordinal alpha → ∀p : set → prop, SNoLev (PSNo alpha p) = alpha
In Proofgold the corresponding term root is 07f1eb... and proposition id is 9c4890...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNo_Subq)
∀x y, SNo x → SNo y → SNoLev x ⊆ SNoLev y → (∀alpha ∈ SNoLev x, alpha ∈ x ↔ alpha ∈ y) → x ⊆ y
In Proofgold the corresponding term root is a244d7... and proposition id is a428af...
Proof:
The rest of this subproof is missing.
∎
Definition. We define SNoEq_ to be λalpha x y ⇒ PNoEq_ alpha (λbeta ⇒ beta ∈ x) (λbeta ⇒ beta ∈ y) of type set → set → set → prop.
In Proofgold the corresponding term root is 5f11e2... and object id is 12b5f6...
Theorem. (SNoEq_I)
∀alpha x y, (∀beta ∈ alpha, beta ∈ x ↔ beta ∈ y) → SNoEq_ alpha x y
In Proofgold the corresponding term root is a718b1... and proposition id is fdcb78...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNo_eq)
∀x y, SNo x → SNo y → SNoLev x = SNoLev y → SNoEq_ (SNoLev x) x y → x = y
In Proofgold the corresponding term root is b08bdc... and proposition id is bf4ac3...
Proof:
The rest of this subproof is missing.
∎
End of Section TaggedSets
Definition. We define SNoLt to be λx y ⇒ PNoLt (SNoLev x) (λbeta ⇒ beta ∈ x) (SNoLev y) (λbeta ⇒ beta ∈ y) of type set → set → prop.
In Proofgold the corresponding term root is 46e7ed... and object id is 8c8aa9...
Notation. We use < as an infix operator with priority 490 and no associativity corresponding to applying term SNoLt.
Definition. We define SNoLe to be λx y ⇒ PNoLe (SNoLev x) (λbeta ⇒ beta ∈ x) (SNoLev y) (λbeta ⇒ beta ∈ y) of type set → set → prop.
In Proofgold the corresponding term root is ddf7d3... and object id is 54f405...
Notation. We use ≤ as an infix operator with priority 490 and no associativity corresponding to applying term SNoLe.
Theorem. (SNoLtLe)
∀x y, x < y → x ≤ y
In Proofgold the corresponding term root is c0fef1... and proposition id is f09833...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNoLeE)
∀x y, SNo x → SNo y → x ≤ y → x < y ∨ x = y
In Proofgold the corresponding term root is 1c9c62... and proposition id is 23cc47...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNoEq_sym_)
∀alpha x y, SNoEq_ alpha x y → SNoEq_ alpha y x
In Proofgold the corresponding term root is f8c61d... and proposition id is f756d2...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNoEq_tra_)
∀alpha x y z, SNoEq_ alpha x y → SNoEq_ alpha y z → SNoEq_ alpha x z
In Proofgold the corresponding term root is 0fff96... and proposition id is 246d04...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNoLtE)
∀x y, SNo x → SNo y → x < y → ∀p : prop, (∀z, SNo z → SNoLev z ∈ SNoLev x ∩ SNoLev y → SNoEq_ (SNoLev z) z x → SNoEq_ (SNoLev z) z y → x < z → z < y → SNoLev z ∉ x → SNoLev z ∈ y → p) → (SNoLev x ∈ SNoLev y → SNoEq_ (SNoLev x) x y → SNoLev x ∈ y → p) → (SNoLev y ∈ SNoLev x → SNoEq_ (SNoLev y) x y → SNoLev y ∉ x → p) → p
In Proofgold the corresponding term root is 958711... and proposition id is d12fc6...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNoLtI2)
∀x y, SNoLev x ∈ SNoLev y → SNoEq_ (SNoLev x) x y → SNoLev x ∈ y → x < y
In Proofgold the corresponding term root is 282b01... and proposition id is 82b82c...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNoLtI3)
∀x y, SNoLev y ∈ SNoLev x → SNoEq_ (SNoLev y) x y → SNoLev y ∉ x → x < y
In Proofgold the corresponding term root is 006afc... and proposition id is 938f67...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNoLt_irref)
∀x, ¬ SNoLt x x
In Proofgold the corresponding term root is d5ecf9... and proposition id is 06fac5...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNoLt_trichotomy_or)
∀x y, SNo x → SNo y → x < y ∨ x = y ∨ y < x
In Proofgold the corresponding term root is 35a4c8... and proposition id is 8ad056...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNoLt_trichotomy_or_impred)
∀x y, SNo x → SNo y → ∀p : prop, (x < y → p) → (x = y → p) → (y < x → p) → p
In Proofgold the corresponding term root is ba04ef... and proposition id is 7588e3...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNoLt_tra)
∀x y z, SNo x → SNo y → SNo z → x < y → y < z → x < z
In Proofgold the corresponding term root is ac0e24... and proposition id is ccfdf7...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNoLe_ref)
∀x, SNoLe x x
In Proofgold the corresponding term root is c940c8... and proposition id is 83bd2d...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNoLe_antisym)
∀x y, SNo x → SNo y → x ≤ y → y ≤ x → x = y
In Proofgold the corresponding term root is f9f757... and proposition id is 3abee4...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNoLtLe_tra)
∀x y z, SNo x → SNo y → SNo z → x < y → y ≤ z → x < z
In Proofgold the corresponding term root is 3bc661... and proposition id is ba8545...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNoLeLt_tra)
∀x y z, SNo x → SNo y → SNo z → x ≤ y → y < z → x < z
In Proofgold the corresponding term root is b2178f... and proposition id is 4a77a2...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNoLe_tra)
∀x y z, SNo x → SNo y → SNo z → x ≤ y → y ≤ z → x ≤ z
In Proofgold the corresponding term root is a0c5c6... and proposition id is 031989...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNoLtLe_or)
∀x y, SNo x → SNo y → x < y ∨ y ≤ x
In Proofgold the corresponding term root is f595ac... and proposition id is 8965d2...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNoLt_PSNo_PNoLt)
∀alpha beta, ∀p q : set → prop, ordinal alpha → ordinal beta → PSNo alpha p < PSNo beta q → PNoLt alpha p beta q
In Proofgold the corresponding term root is 6e122d... and proposition id is 535161...
Proof:
The rest of this subproof is missing.
∎
Theorem. (PNoLt_SNoLt_PSNo)
∀alpha beta, ∀p q : set → prop, ordinal alpha → ordinal beta → PNoLt alpha p beta q → PSNo alpha p < PSNo beta q
In Proofgold the corresponding term root is 11b6ed... and proposition id is 6196d9...
Proof:
The rest of this subproof is missing.
∎
Definition. We define SNoCut to be λL R ⇒ PSNo (PNo_bd (λalpha p ⇒ ordinal alpha ∧ PSNo alpha p ∈ L) (λalpha p ⇒ ordinal alpha ∧ PSNo alpha p ∈ R)) (PNo_pred (λalpha p ⇒ ordinal alpha ∧ PSNo alpha p ∈ L) (λalpha p ⇒ ordinal alpha ∧ PSNo alpha p ∈ R)) of type set → set → set.
In Proofgold the corresponding term root is ec849e... and object id is 295bc4...
Definition. We define SNoCutP to be λL R ⇒ (∀x ∈ L, SNo x) ∧ (∀y ∈ R, SNo y) ∧ (∀x ∈ L, ∀y ∈ R, x < y) of type set → set → prop.
In Proofgold the corresponding term root is c083d8... and object id is 775127...
Theorem. (SNoCutP_SNoCut)
∀L R, SNoCutP L R → SNo (SNoCut L R) ∧ SNoLev (SNoCut L R) ∈ ordsucc ((⋃x ∈ Lordsucc (SNoLev x)) ∪ (⋃y ∈ Rordsucc (SNoLev y))) ∧ (∀x ∈ L, x < SNoCut L R) ∧ (∀y ∈ R, SNoCut L R < y) ∧ (∀z, SNo z → (∀x ∈ L, x < z) → (∀y ∈ R, z < y) → SNoLev (SNoCut L R) ⊆ SNoLev z ∧ SNoEq_ (SNoLev (SNoCut L R)) (SNoCut L R) z)
In Proofgold the corresponding term root is 9398b7... and proposition id is 66456c...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNoCutP_SNoCut_impred)
∀L R, SNoCutP L R → ∀p : prop, (SNo (SNoCut L R) → SNoLev (SNoCut L R) ∈ ordsucc ((⋃x ∈ Lordsucc (SNoLev x)) ∪ (⋃y ∈ Rordsucc (SNoLev y))) → (∀x ∈ L, x < SNoCut L R) → (∀y ∈ R, SNoCut L R < y) → (∀z, SNo z → (∀x ∈ L, x < z) → (∀y ∈ R, z < y) → SNoLev (SNoCut L R) ⊆ SNoLev z ∧ SNoEq_ (SNoLev (SNoCut L R)) (SNoCut L R) z) → p) → p
In Proofgold the corresponding term root is 02c4ae... and proposition id is 5bcc8f...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNoCutP_L_0)
∀L, (∀x ∈ L, SNo x) → SNoCutP L 0
In Proofgold the corresponding term root is 47329c... and proposition id is ba61e1...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNoCutP_0_0)
SNoCutP 0 0
In Proofgold the corresponding term root is 3b144e... and proposition id is a5144d...
Proof:
The rest of this subproof is missing.
∎
Definition. We define SNoS_ to be λalpha ⇒ {x ∈ 𝒫 (SNoElts_ alpha)|∃beta ∈ alpha, SNo_ beta x} of type set → set.
In Proofgold the corresponding term root is d5069a... and object id is 4680ce...
Theorem. (SNoS_E)
∀alpha, ordinal alpha → ∀x ∈ SNoS_ alpha, ∃beta ∈ alpha, SNo_ beta x
In Proofgold the corresponding term root is 970ce2... and proposition id is bb731c...
Proof:
The rest of this subproof is missing.
∎
Beginning of Section TaggedSets2
Let tag : set → set ≝ λalpha ⇒ SetAdjoin alpha {1}
Notation. We use ' as a postfix operator with priority 100 corresponding to applying term tag.
Theorem. (SNoS_I)
∀alpha, ordinal alpha → ∀x, ∀beta ∈ alpha, SNo_ beta x → x ∈ SNoS_ alpha
In Proofgold the corresponding term root is f5227d... and proposition id is 04f24b...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNoS_I2)
∀x y, SNo x → SNo y → SNoLev x ∈ SNoLev y → x ∈ SNoS_ (SNoLev y)
In Proofgold the corresponding term root is 7a14c7... and proposition id is b4042d...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNoS_Subq)
∀alpha beta, ordinal alpha → ordinal beta → alpha ⊆ beta → SNoS_ alpha ⊆ SNoS_ beta
In Proofgold the corresponding term root is 6828e7... and proposition id is d928b8...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNoLev_uniq2)
∀alpha, ordinal alpha → ∀x, SNo_ alpha x → SNoLev x = alpha
In Proofgold the corresponding term root is fbb461... and proposition id is f48d47...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNoS_E2)
∀alpha, ordinal alpha → ∀x ∈ SNoS_ alpha, ∀p : prop, (SNoLev x ∈ alpha → ordinal (SNoLev x) → SNo x → SNo_ (SNoLev x) x → p) → p
In Proofgold the corresponding term root is deb031... and proposition id is 62b490...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNoS_In_neq)
∀w, SNo w → ∀x ∈ SNoS_ (SNoLev w), x ≠ w
In Proofgold the corresponding term root is ab85fd... and proposition id is d5dbd7...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNoS_SNoLev)
∀z, SNo z → z ∈ SNoS_ (ordsucc (SNoLev z))
In Proofgold the corresponding term root is a9ba1f... and proposition id is 8efb88...
Proof:
The rest of this subproof is missing.
∎
Definition. We define SNoL to be λz ⇒ {x ∈ SNoS_ (SNoLev z)|x < z} of type set → set.
In Proofgold the corresponding term root is 8cd812... and object id is 48bcf8...
Definition. We define SNoR to be λz ⇒ {y ∈ SNoS_ (SNoLev z)|z < y} of type set → set.
In Proofgold the corresponding term root is 73b2b9... and object id is 0cfe87...
Theorem. (SNoCutP_SNoL_SNoR)
∀z, SNo z → SNoCutP (SNoL z) (SNoR z)
In Proofgold the corresponding term root is 524866... and proposition id is 7aba57...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNoL_E)
∀x, SNo x → ∀w ∈ SNoL x, ∀p : prop, (SNo w → SNoLev w ∈ SNoLev x → w < x → p) → p
In Proofgold the corresponding term root is cead4f... and proposition id is 7ccd8b...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNoR_E)
∀x, SNo x → ∀z ∈ SNoR x, ∀p : prop, (SNo z → SNoLev z ∈ SNoLev x → x < z → p) → p
In Proofgold the corresponding term root is efca05... and proposition id is 7dca04...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNoL_SNoS_)
∀z, SNoL z ⊆ SNoS_ (SNoLev z)
In Proofgold the corresponding term root is 73f5b2... and proposition id is 409713...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNoR_SNoS_)
∀z, SNoR z ⊆ SNoS_ (SNoLev z)
In Proofgold the corresponding term root is 16bdfc... and proposition id is 2fe1c0...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNoL_SNoS)
∀x, SNo x → ∀w ∈ SNoL x, w ∈ SNoS_ (SNoLev x)
In Proofgold the corresponding term root is 6e3a22... and proposition id is ded78a...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNoR_SNoS)
∀x, SNo x → ∀z ∈ SNoR x, z ∈ SNoS_ (SNoLev x)
In Proofgold the corresponding term root is 98051c... and proposition id is e13f49...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNoL_I)
∀z, SNo z → ∀x, SNo x → SNoLev x ∈ SNoLev z → x < z → x ∈ SNoL z
In Proofgold the corresponding term root is ddd91b... and proposition id is 4f3d04...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNoR_I)
∀z, SNo z → ∀y, SNo y → SNoLev y ∈ SNoLev z → z < y → y ∈ SNoR z
In Proofgold the corresponding term root is 97693c... and proposition id is 383e88...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNo_eta)
∀z, SNo z → z = SNoCut (SNoL z) (SNoR z)
In Proofgold the corresponding term root is 3cc369... and proposition id is dc9273...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNoCutP_SNo_SNoCut)
∀L R, SNoCutP L R → SNo (SNoCut L R)
In Proofgold the corresponding term root is 9621aa... and proposition id is ae46e8...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNoCutP_SNoCut_L)
∀L R, SNoCutP L R → ∀x ∈ L, x < SNoCut L R
In Proofgold the corresponding term root is 984a30... and proposition id is 4bca14...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNoCutP_SNoCut_R)
∀L R, SNoCutP L R → ∀y ∈ R, SNoCut L R < y
In Proofgold the corresponding term root is 9161db... and proposition id is 7c3d12...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNoCutP_SNoCut_fst)
∀L R, SNoCutP L R → ∀z, SNo z → (∀x ∈ L, x < z) → (∀y ∈ R, z < y) → SNoLev (SNoCut L R) ⊆ SNoLev z ∧ SNoEq_ (SNoLev (SNoCut L R)) (SNoCut L R) z
In Proofgold the corresponding term root is b52eb7... and proposition id is 8362f8...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNoCut_Le)
∀L1 R1 L2 R2, SNoCutP L1 R1 → SNoCutP L2 R2 → (∀w ∈ L1, w < SNoCut L2 R2) → (∀z ∈ R2, SNoCut L1 R1 < z) → SNoCut L1 R1 ≤ SNoCut L2 R2
In Proofgold the corresponding term root is bb8cdc... and proposition id is 423731...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNoCut_ext)
∀L1 R1 L2 R2, SNoCutP L1 R1 → SNoCutP L2 R2 → (∀w ∈ L1, w < SNoCut L2 R2) → (∀z ∈ R1, SNoCut L2 R2 < z) → (∀w ∈ L2, w < SNoCut L1 R1) → (∀z ∈ R2, SNoCut L1 R1 < z) → SNoCut L1 R1 = SNoCut L2 R2
In Proofgold the corresponding term root is c1078e... and proposition id is f66618...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNoLt_SNoL_or_SNoR_impred)
∀x y, SNo x → SNo y → x < y → ∀p : prop, (∀z ∈ SNoL y, z ∈ SNoR x → p) → (x ∈ SNoL y → p) → (y ∈ SNoR x → p) → p
In Proofgold the corresponding term root is f3b15a... and proposition id is ffc0bf...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNoL_or_SNoR_impred)
∀x y, SNo x → SNo y → ∀p : prop, (x = y → p) → (∀z ∈ SNoL y, z ∈ SNoR x → p) → (x ∈ SNoL y → p) → (y ∈ SNoR x → p) → (∀z ∈ SNoR y, z ∈ SNoL x → p) → (x ∈ SNoR y → p) → (y ∈ SNoL x → p) → p
In Proofgold the corresponding term root is f8a5e2... and proposition id is 68ba98...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNoL_SNoCutP_ex)
∀L R, SNoCutP L R → ∀w ∈ SNoL (SNoCut L R), ∃w' ∈ L, w ≤ w'
In Proofgold the corresponding term root is 3b4695... and proposition id is 6d8af6...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNoR_SNoCutP_ex)
∀L R, SNoCutP L R → ∀z ∈ SNoR (SNoCut L R), ∃z' ∈ R, z' ≤ z
In Proofgold the corresponding term root is ca7be2... and proposition id is bb0b53...
Proof:
The rest of this subproof is missing.
∎
Theorem. (ordinal_SNo_)
∀alpha, ordinal alpha → SNo_ alpha alpha
In Proofgold the corresponding term root is 4731fd... and proposition id is 7754e5...
Proof:
The rest of this subproof is missing.
∎
Theorem. (ordinal_SNo)
∀alpha, ordinal alpha → SNo alpha
In Proofgold the corresponding term root is ac39bc... and proposition id is 1eea33...
Proof:
The rest of this subproof is missing.
∎
Theorem. (ordinal_SNoLev)
∀alpha, ordinal alpha → SNoLev alpha = alpha
In Proofgold the corresponding term root is 56f8b0... and proposition id is fea5fc...
Proof:
The rest of this subproof is missing.
∎
Theorem. (ordinal_SNoLev_max)
∀alpha, ordinal alpha → ∀z, SNo z → SNoLev z ∈ alpha → z < alpha
In Proofgold the corresponding term root is 945794... and proposition id is 51ee47...
Proof:
The rest of this subproof is missing.
∎
Theorem. (ordinal_SNoL)
∀alpha, ordinal alpha → SNoL alpha = SNoS_ alpha
In Proofgold the corresponding term root is fef608... and proposition id is 95823e...
Proof:
The rest of this subproof is missing.
∎
Theorem. (ordinal_SNoR)
∀alpha, ordinal alpha → SNoR alpha = Empty
In Proofgold the corresponding term root is 07c69a... and proposition id is f853d9...
Proof:
The rest of this subproof is missing.
∎
Theorem. (nat_p_SNo)
∀n, nat_p n → SNo n
In Proofgold the corresponding term root is 0644f3... and proposition id is 5fd930...
Proof:
The rest of this subproof is missing.
∎
Theorem. (omega_SNo)
∀n ∈ ω, SNo n
In Proofgold the corresponding term root is d32269... and proposition id is d0f9c6...
Proof:
The rest of this subproof is missing.
∎
Theorem. (omega_SNoS_omega)
ω ⊆ SNoS_ ω
In Proofgold the corresponding term root is 4ba265... and proposition id is 2ccf7a...
Proof:
The rest of this subproof is missing.
∎
Theorem. (ordinal_In_SNoLt)
∀alpha, ordinal alpha → ∀beta ∈ alpha, beta < alpha
In Proofgold the corresponding term root is e37d00... and proposition id is ced78e...
Proof:
The rest of this subproof is missing.
∎
Theorem. (ordinal_SNoLev_max_2)
∀alpha, ordinal alpha → ∀z, SNo z → SNoLev z ∈ ordsucc alpha → z ≤ alpha
In Proofgold the corresponding term root is 72c5eb... and proposition id is c9a731...
Proof:
The rest of this subproof is missing.
∎
Theorem. (ordinal_Subq_SNoLe)
∀alpha beta, ordinal alpha → ordinal beta → alpha ⊆ beta → alpha ≤ beta
In Proofgold the corresponding term root is 7558ba... and proposition id is 15c58a...
Proof:
The rest of this subproof is missing.
∎
Theorem. (ordinal_SNoLt_In)
∀alpha beta, ordinal alpha → ordinal beta → alpha < beta → alpha ∈ beta
In Proofgold the corresponding term root is d6f066... and proposition id is de775b...
Proof:
The rest of this subproof is missing.
∎
Theorem. (omega_nonneg)
∀m ∈ ω, 0 ≤ m
In Proofgold the corresponding term root is ee67c9... and proposition id is eaf06b...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNo_0)
SNo 0
In Proofgold the corresponding term root is c8204b... and proposition id is 94a310...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNo_1)
SNo 1
In Proofgold the corresponding term root is ac50d7... and proposition id is 9a733a...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNo_2)
SNo 2
In Proofgold the corresponding term root is 487591... and proposition id is c524a4...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNoLev_0)
SNoLev 0 = 0
In Proofgold the corresponding term root is 45a0f7... and proposition id is 3527a4...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNoCut_0_0)
SNoCut 0 0 = 0
In Proofgold the corresponding term root is 6a1f81... and proposition id is abbffb...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNoL_0)
SNoL 0 = 0
In Proofgold the corresponding term root is c01c2b... and proposition id is fdaa69...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNoR_0)
SNoR 0 = 0
In Proofgold the corresponding term root is 7329c7... and proposition id is 510751...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNoL_1)
SNoL 1 = 1
In Proofgold the corresponding term root is 84d688... and proposition id is 15aa79...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNoR_1)
SNoR 1 = 0
In Proofgold the corresponding term root is e6a83a... and proposition id is 8da427...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNo_max_SNoLev)
∀x, SNo x → (∀y ∈ SNoS_ (SNoLev x), y < x) → SNoLev x = x
In Proofgold the corresponding term root is ff019e... and proposition id is 3be246...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNo_max_ordinal)
∀x, SNo x → (∀y ∈ SNoS_ (SNoLev x), y < x) → ordinal x
In Proofgold the corresponding term root is 0b230c... and proposition id is f28b9a...
Proof:
The rest of this subproof is missing.
∎
Theorem. (pos_low_eq_one)
∀x, SNo x → 0 < x → SNoLev x ⊆ 1 → x = 1
In Proofgold the corresponding term root is 7c84ef... and proposition id is db40ad...
Proof:
The rest of this subproof is missing.
∎
Definition. We define SNo_extend0 to be λx ⇒ PSNo (ordsucc (SNoLev x)) (λdelta ⇒ delta ∈ x ∧ delta ≠ SNoLev x) of type set → set.
In Proofgold the corresponding term root is 997d91... and object id is ff4161...
Definition. We define SNo_extend1 to be λx ⇒ PSNo (ordsucc (SNoLev x)) (λdelta ⇒ delta ∈ x ∨ delta = SNoLev x) of type set → set.
In Proofgold the corresponding term root is 464e47... and object id is 8c551b...
Theorem. (SNo_extend0_SNo_)
∀x, SNo x → SNo_ (ordsucc (SNoLev x)) (SNo_extend0 x)
In Proofgold the corresponding term root is d41d3e... and proposition id is c8f8ea...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNo_extend1_SNo_)
∀x, SNo x → SNo_ (ordsucc (SNoLev x)) (SNo_extend1 x)
In Proofgold the corresponding term root is b7b866... and proposition id is 73bbf2...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNo_extend0_SNo)
∀x, SNo x → SNo (SNo_extend0 x)
In Proofgold the corresponding term root is b353a2... and proposition id is 76562c...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNo_extend1_SNo)
∀x, SNo x → SNo (SNo_extend1 x)
In Proofgold the corresponding term root is fa323a... and proposition id is 5345a7...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNo_extend0_SNoLev)
∀x, SNo x → SNoLev (SNo_extend0 x) = ordsucc (SNoLev x)
In Proofgold the corresponding term root is 26efcd... and proposition id is 5aa0af...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNo_extend1_SNoLev)
∀x, SNo x → SNoLev (SNo_extend1 x) = ordsucc (SNoLev x)
In Proofgold the corresponding term root is 33cf60... and proposition id is 529347...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNo_extend0_nIn)
∀x, SNo x → SNoLev x ∉ SNo_extend0 x
In Proofgold the corresponding term root is 217cff... and proposition id is 0dfd17...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNo_extend1_In)
∀x, SNo x → SNoLev x ∈ SNo_extend1 x
In Proofgold the corresponding term root is ac9570... and proposition id is 8ca372...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNo_extend0_SNoEq)
∀x, SNo x → SNoEq_ (SNoLev x) (SNo_extend0 x) x
In Proofgold the corresponding term root is bd4a27... and proposition id is 129986...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNo_extend1_SNoEq)
∀x, SNo x → SNoEq_ (SNoLev x) (SNo_extend1 x) x
In Proofgold the corresponding term root is ba974f... and proposition id is d6f8bb...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNoLev_0_eq_0)
∀x, SNo x → SNoLev x = 0 → x = 0
In Proofgold the corresponding term root is fe0bc4... and proposition id is dfcf39...
Proof:
The rest of this subproof is missing.
∎
Definition. We define eps_ to be λn ⇒ {0} ∪ {(ordsucc m) '|m ∈ n} of type set → set.
In Proofgold the corresponding term root is 5e992a... and object id is ffee2b...
Theorem. (eps_ordinal_In_eq_0)
∀n alpha, ordinal alpha → alpha ∈ eps_ n → alpha = 0
In Proofgold the corresponding term root is 3046fd... and proposition id is 1190f3...
Proof:
The rest of this subproof is missing.
∎
Theorem. (eps_0_1)
eps_ 0 = 1
In Proofgold the corresponding term root is 5bbce1... and proposition id is d073f0...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNo__eps_)
∀n ∈ ω, SNo_ (ordsucc n) (eps_ n)
In Proofgold the corresponding term root is e9c4d9... and proposition id is 1c1428...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNo_eps_)
∀n ∈ ω, SNo (eps_ n)
In Proofgold the corresponding term root is 9fb700... and proposition id is 006b0f...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNo_eps_1)
SNo (eps_ 1)
In Proofgold the corresponding term root is 17c276... and proposition id is 8b78d7...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNoLev_eps_)
∀n ∈ ω, SNoLev (eps_ n) = ordsucc n
In Proofgold the corresponding term root is 6891e3... and proposition id is 75efd1...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNo_eps_SNoS_omega)
∀n ∈ ω, eps_ n ∈ SNoS_ ω
In Proofgold the corresponding term root is 736d2a... and proposition id is 52da18...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNo_eps_decr)
∀n ∈ ω, ∀m ∈ n, eps_ n < eps_ m
In Proofgold the corresponding term root is d6a46a... and proposition id is 67c76a...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNo_eps_pos)
∀n ∈ ω, 0 < eps_ n
In Proofgold the corresponding term root is 8b75de... and proposition id is 760b4a...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNo_pos_eps_Lt)
∀n, nat_p n → ∀x ∈ SNoS_ (ordsucc n), 0 < x → eps_ n < x
In Proofgold the corresponding term root is 72974f... and proposition id is 771b0e...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNo_pos_eps_Le)
∀n, nat_p n → ∀x ∈ SNoS_ (ordsucc (ordsucc n)), 0 < x → eps_ n ≤ x
In Proofgold the corresponding term root is f1e297... and proposition id is 6658de...
Proof:
The rest of this subproof is missing.
∎
Theorem. (eps_SNo_eq)
∀n, nat_p n → ∀x ∈ SNoS_ (ordsucc n), 0 < x → SNoEq_ (SNoLev x) (eps_ n) x → ∃m ∈ n, x = eps_ m
In Proofgold the corresponding term root is 8c2587... and proposition id is 8515ac...
Proof:
The rest of this subproof is missing.
∎
Theorem. (eps_SNoCutP)
∀n ∈ ω, SNoCutP {0} {eps_ m|m ∈ n}
In Proofgold the corresponding term root is 545873... and proposition id is 30edf9...
Proof:
The rest of this subproof is missing.
∎
Theorem. (eps_SNoCut)
∀n ∈ ω, eps_ n = SNoCut {0} {eps_ m|m ∈ n}
In Proofgold the corresponding term root is 06d6c5... and proposition id is eba0c6...
Proof:
The rest of this subproof is missing.
∎
End of Section TaggedSets2
Theorem. (SNo_etaE)
∀z, SNo z → ∀p : prop, (∀L R, SNoCutP L R → (∀x ∈ L, SNoLev x ∈ SNoLev z) → (∀y ∈ R, SNoLev y ∈ SNoLev z) → z = SNoCut L R → p) → p
In Proofgold the corresponding term root is c24473... and proposition id is 109f1d...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNo_ind)
∀P : set → prop, (∀L R, SNoCutP L R → (∀x ∈ L, P x) → (∀y ∈ R, P y) → P (SNoCut L R)) → ∀z, SNo z → P z
In Proofgold the corresponding term root is 5b4c39... and proposition id is 94cbe6...
Proof:
The rest of this subproof is missing.
∎
Beginning of Section SurrealRecI
Variable F : set → (set → set) → set
Let default : set ≝ Eps_i (λ_ ⇒ True)
Let G : set → (set → set → set) → set → set ≝ λalpha g ⇒ If_ii (ordinal alpha) (λz : set ⇒ if z ∈ SNoS_ (ordsucc alpha) then F z (λw ⇒ g (SNoLev w) w) else default) (λz : set ⇒ default)
Definition. We define SNo_rec_i to be λz ⇒ In_rec_ii G (SNoLev z) z of type set → set.
In Proofgold the corresponding term root is be45df... and object id is 230ba8...
Hypothesis Fr : ∀z, SNo z → ∀g h : set → set, (∀w ∈ SNoS_ (SNoLev z), g w = h w) → F z g = F z h
Theorem. (SNo_rec_i_eq)
∀z, SNo z → SNo_rec_i z = F z SNo_rec_i
In Proofgold the corresponding term root is d5ffa8... and proposition id is f547c8...
Proof:
The rest of this subproof is missing.
∎
End of Section SurrealRecI
Beginning of Section SurrealRecII
Variable F : set → (set → (set → set)) → (set → set)
Let default : (set → set) ≝ Descr_ii (λ_ ⇒ True)
Let G : set → (set → set → (set → set)) → set → (set → set) ≝ λalpha g ⇒ If_iii (ordinal alpha) (λz : set ⇒ If_ii (z ∈ SNoS_ (ordsucc alpha)) (F z (λw ⇒ g (SNoLev w) w)) default) (λz : set ⇒ default)
Definition. We define SNo_rec_ii to be λz ⇒ In_rec_iii G (SNoLev z) z of type set → (set → set).
In Proofgold the corresponding term root is e148e6... and object id is c3f218...
Hypothesis Fr : ∀z, SNo z → ∀g h : set → (set → set), (∀w ∈ SNoS_ (SNoLev z), g w = h w) → F z g = F z h
Theorem. (SNo_rec_ii_eq)
∀z, SNo z → SNo_rec_ii z = F z SNo_rec_ii
In Proofgold the corresponding term root is 39e866... and proposition id is 1c5373...
Proof:
The rest of this subproof is missing.
∎
End of Section SurrealRecII
Beginning of Section SurrealRec2
Variable F : set → set → (set → set → set) → set
Let G : set → (set → set → set) → set → (set → set) → set ≝ λw f z g ⇒ F w z (λx y ⇒ if x = w then g y else f x y)
Let H : set → (set → set → set) → set → set ≝ λw f z ⇒ if SNo z then SNo_rec_i (G w f) z else Empty
Definition. We define SNo_rec2 to be SNo_rec_ii H of type set → set → set.
In Proofgold the corresponding term root is 7d10ab... and object id is 951110...
Hypothesis Fr : ∀w, SNo w → ∀z, SNo z → ∀g h : set → set → set, (∀x ∈ SNoS_ (SNoLev w), ∀y, SNo y → g x y = h x y) → (∀y ∈ SNoS_ (SNoLev z), g w y = h w y) → F w z g = F w z h
Theorem. (SNo_rec2_G_prop)
∀w, SNo w → ∀f k : set → set → set, (∀x ∈ SNoS_ (SNoLev w), f x = k x) → ∀z, SNo z → ∀g h : set → set, (∀u ∈ SNoS_ (SNoLev z), g u = h u) → G w f z g = G w k z h
In Proofgold the corresponding term root is 0f875d... and proposition id is de8737...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNo_rec2_eq_1)
∀w, SNo w → ∀f : set → set → set, ∀z, SNo z → SNo_rec_i (G w f) z = G w f z (SNo_rec_i (G w f))
In Proofgold the corresponding term root is 12a51a... and proposition id is 0dc50e...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNo_rec2_eq)
∀w, SNo w → ∀z, SNo z → SNo_rec2 w z = F w z SNo_rec2
In Proofgold the corresponding term root is e9ea41... and proposition id is ac9f2b...
Proof:
The rest of this subproof is missing.
∎
End of Section SurrealRec2
Theorem. (SNo_ordinal_ind)
∀P : set → prop, (∀alpha, ordinal alpha → ∀x ∈ SNoS_ alpha, P x) → (∀x, SNo x → P x)
In Proofgold the corresponding term root is 5b75fb... and proposition id is dc2c6d...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNo_ordinal_ind2)
∀P : set → set → prop, (∀alpha, ordinal alpha → ∀beta, ordinal beta → ∀x ∈ SNoS_ alpha, ∀y ∈ SNoS_ beta, P x y) → (∀x y, SNo x → SNo y → P x y)
In Proofgold the corresponding term root is ba3315... and proposition id is 4cd5ac...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNo_ordinal_ind3)
∀P : set → set → set → prop, (∀alpha, ordinal alpha → ∀beta, ordinal beta → ∀gamma, ordinal gamma → ∀x ∈ SNoS_ alpha, ∀y ∈ SNoS_ beta, ∀z ∈ SNoS_ gamma, P x y z) → (∀x y z, SNo x → SNo y → SNo z → P x y z)
In Proofgold the corresponding term root is fccdb6... and proposition id is dece8c...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNoLev_ind)
∀P : set → prop, (∀x, SNo x → (∀w ∈ SNoS_ (SNoLev x), P w) → P x) → (∀x, SNo x → P x)
In Proofgold the corresponding term root is 8b2b42... and proposition id is 851a88...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNoLev_ind2)
∀P : set → set → prop, (∀x y, SNo x → SNo y → (∀w ∈ SNoS_ (SNoLev x), P w y) → (∀z ∈ SNoS_ (SNoLev y), P x z) → (∀w ∈ SNoS_ (SNoLev x), ∀z ∈ SNoS_ (SNoLev y), P w z) → P x y) → ∀x y, SNo x → SNo y → P x y
In Proofgold the corresponding term root is 87eb9a... and proposition id is 800d4c...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNoLev_ind3)
∀P : set → set → set → prop, (∀x y z, SNo x → SNo y → SNo z → (∀u ∈ SNoS_ (SNoLev x), P u y z) → (∀v ∈ SNoS_ (SNoLev y), P x v z) → (∀w ∈ SNoS_ (SNoLev z), P x y w) → (∀u ∈ SNoS_ (SNoLev x), ∀v ∈ SNoS_ (SNoLev y), P u v z) → (∀u ∈ SNoS_ (SNoLev x), ∀w ∈ SNoS_ (SNoLev z), P u y w) → (∀v ∈ SNoS_ (SNoLev y), ∀w ∈ SNoS_ (SNoLev z), P x v w) → (∀u ∈ SNoS_ (SNoLev x), ∀v ∈ SNoS_ (SNoLev y), ∀w ∈ SNoS_ (SNoLev z), P u v w) → P x y z) → ∀x y z, SNo x → SNo y → SNo z → P x y z
In Proofgold the corresponding term root is f1d779... and proposition id is ee16b9...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNo_omega)
SNo ω
In Proofgold the corresponding term root is f65848... and proposition id is db7b5f...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNoLt_0_1)
0 < 1
In Proofgold the corresponding term root is dd6047... and proposition id is cb6392...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNoLt_0_2)
0 < 2
In Proofgold the corresponding term root is 7be36f... and proposition id is f46934...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNoLt_1_2)
1 < 2
In Proofgold the corresponding term root is cc44e8... and proposition id is 19d5e1...
Proof:
The rest of this subproof is missing.
∎
Theorem. (restr_SNo_)
∀x, SNo x → ∀alpha ∈ SNoLev x, SNo_ alpha (x ∩ SNoElts_ alpha)
In Proofgold the corresponding term root is d78444... and proposition id is e0eca8...
Proof:
The rest of this subproof is missing.
∎
Theorem. (restr_SNo)
∀x, SNo x → ∀alpha ∈ SNoLev x, SNo (x ∩ SNoElts_ alpha)
In Proofgold the corresponding term root is 930b56... and proposition id is d141ed...
Proof:
The rest of this subproof is missing.
∎
Theorem. (restr_SNoLev)
∀x, SNo x → ∀alpha ∈ SNoLev x, SNoLev (x ∩ SNoElts_ alpha) = alpha
In Proofgold the corresponding term root is cae8b5... and proposition id is 8b1853...
Proof:
The rest of this subproof is missing.
∎
Theorem. (restr_SNoEq)
∀x, SNo x → ∀alpha ∈ SNoLev x, SNoEq_ alpha (x ∩ SNoElts_ alpha) x
In Proofgold the corresponding term root is 829f8b... and proposition id is 704f5a...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNo_extend0_restr_eq)
∀x, SNo x → x = SNo_extend0 x ∩ SNoElts_ (SNoLev x)
In Proofgold the corresponding term root is 2c202e... and proposition id is 9827a9...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNo_extend1_restr_eq)
∀x, SNo x → x = SNo_extend1 x ∩ SNoElts_ (SNoLev x)
In Proofgold the corresponding term root is 75bbe8... and proposition id is 275196...
Proof:
The rest of this subproof is missing.
∎
Beginning of Section SurrealMinus
Definition. We define minus_SNo to be SNo_rec_i (λx m ⇒ SNoCut {m z|z ∈ SNoR x} {m w|w ∈ SNoL x}) of type set → set.
In Proofgold the corresponding term root is 268a6c... and object id is 8a6d85...
Notation. We use - as a prefix operator with priority 358 corresponding to applying term minus_SNo.
Notation. We use ≤ as an infix operator with priority 490 and no associativity corresponding to applying term SNoLe.
Theorem. (minus_SNo_eq)
∀x, SNo x → - x = SNoCut {- z|z ∈ SNoR x} {- w|w ∈ SNoL x}
In Proofgold the corresponding term root is c9793e... and proposition id is 570cba...
Proof:
The rest of this subproof is missing.
∎
Theorem. (minus_SNo_prop1)
∀x, SNo x → SNo (- x) ∧ (∀u ∈ SNoL x, - x < - u) ∧ (∀u ∈ SNoR x, - u < - x) ∧ SNoCutP {- z|z ∈ SNoR x} {- w|w ∈ SNoL x}
In Proofgold the corresponding term root is 7c0e82... and proposition id is f0b75f...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNo_minus_SNo)
∀x, SNo x → SNo (- x)
In Proofgold the corresponding term root is 5cf7d9... and proposition id is 744cc2...
Proof:
The rest of this subproof is missing.
∎
Theorem. (minus_SNo_Lt_contra)
∀x y, SNo x → SNo y → x < y → - y < - x
In Proofgold the corresponding term root is 5ce52d... and proposition id is 3ebf6e...
Proof:
The rest of this subproof is missing.
∎
Theorem. (minus_SNo_Le_contra)
∀x y, SNo x → SNo y → x ≤ y → - y ≤ - x
In Proofgold the corresponding term root is c1b225... and proposition id is f8c896...
Proof:
The rest of this subproof is missing.
∎
Theorem. (minus_SNo_SNoCutP)
∀x, SNo x → SNoCutP {- z|z ∈ SNoR x} {- w|w ∈ SNoL x}
In Proofgold the corresponding term root is 99c6fc... and proposition id is 5ad5e4...
Proof:
The rest of this subproof is missing.
∎
Theorem. (minus_SNo_SNoCutP_gen)
∀L R, SNoCutP L R → SNoCutP {- z|z ∈ R} {- w|w ∈ L}
In Proofgold the corresponding term root is e7650d... and proposition id is 687fea...
Proof:
The rest of this subproof is missing.
∎
Theorem. (minus_SNo_Lev_lem1)
∀alpha, ordinal alpha → ∀x ∈ SNoS_ alpha, SNoLev (- x) ⊆ SNoLev x
In Proofgold the corresponding term root is 14986a... and proposition id is afc115...
Proof:
The rest of this subproof is missing.
∎
Theorem. (minus_SNo_Lev_lem2)
∀x, SNo x → SNoLev (- x) ⊆ SNoLev x
In Proofgold the corresponding term root is 879020... and proposition id is 19cf80...
Proof:
The rest of this subproof is missing.
∎
Theorem. (minus_SNo_invol)
∀x, SNo x → - - x = x
In Proofgold the corresponding term root is 44bfe8... and proposition id is cc64ee...
Proof:
The rest of this subproof is missing.
∎
Theorem. (minus_SNo_Lev)
∀x, SNo x → SNoLev (- x) = SNoLev x
In Proofgold the corresponding term root is f65499... and proposition id is 476ae5...
Proof:
The rest of this subproof is missing.
∎
Theorem. (minus_SNo_SNo_)
∀alpha, ordinal alpha → ∀x, SNo_ alpha x → SNo_ alpha (- x)
In Proofgold the corresponding term root is 84605f... and proposition id is 14bd6d...
Proof:
The rest of this subproof is missing.
∎
Theorem. (minus_SNo_SNoS_)
∀alpha, ordinal alpha → ∀x, x ∈ SNoS_ alpha → - x ∈ SNoS_ alpha
In Proofgold the corresponding term root is 12f1a5... and proposition id is e10c92...
Proof:
The rest of this subproof is missing.
∎
Theorem. (minus_SNoCut_eq_lem)
∀v, SNo v → ∀L R, SNoCutP L R → v = SNoCut L R → - v = SNoCut {- z|z ∈ R} {- w|w ∈ L}
In Proofgold the corresponding term root is 29069a... and proposition id is 11eb03...
Proof:
The rest of this subproof is missing.
∎
Theorem. (minus_SNoCut_eq)
∀L R, SNoCutP L R → - SNoCut L R = SNoCut {- z|z ∈ R} {- w|w ∈ L}
In Proofgold the corresponding term root is 083ffe... and proposition id is 54736f...
Proof:
The rest of this subproof is missing.
∎
Theorem. (minus_SNo_Lt_contra1)
∀x y, SNo x → SNo y → - x < y → - y < x
In Proofgold the corresponding term root is 226757... and proposition id is 298729...
Proof:
The rest of this subproof is missing.
∎
Theorem. (minus_SNo_Lt_contra2)
∀x y, SNo x → SNo y → x < - y → y < - x
In Proofgold the corresponding term root is c6f2da... and proposition id is f41968...
Proof:
The rest of this subproof is missing.
∎
Theorem. (mordinal_SNoLev_min_2)
∀alpha, ordinal alpha → ∀z, SNo z → SNoLev z ∈ ordsucc alpha → - alpha ≤ z
In Proofgold the corresponding term root is 2766c7... and proposition id is a96d68...
Proof:
The rest of this subproof is missing.
∎
Theorem. (minus_SNo_SNoS_omega)
∀x ∈ SNoS_ ω, - x ∈ SNoS_ ω
In Proofgold the corresponding term root is daeba5... and proposition id is 0c19f5...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNoL_minus_SNoR)
∀x, SNo x → SNoL (- x) = {- w|w ∈ SNoR x}
In Proofgold the corresponding term root is 084b2d... and proposition id is 8b2b92...
Proof:
The rest of this subproof is missing.
∎
End of Section SurrealMinus
Beginning of Section SurrealAdd
Notation. We use - as a prefix operator with priority 358 corresponding to applying term minus_SNo.
Definition. We define add_SNo to be SNo_rec2 (λx y a ⇒ SNoCut ({a w y|w ∈ SNoL x} ∪ {a x w|w ∈ SNoL y}) ({a z y|z ∈ SNoR x} ∪ {a x z|z ∈ SNoR y})) of type set → set → set.
In Proofgold the corresponding term root is 127d04... and object id is f62f9a...
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_SNo.
Theorem. (add_SNo_eq)
∀x, SNo x → ∀y, SNo y → x + y = SNoCut ({w + y|w ∈ SNoL x} ∪ {x + w|w ∈ SNoL y}) ({z + y|z ∈ SNoR x} ∪ {x + z|z ∈ SNoR y})
In Proofgold the corresponding term root is 0c1128... and proposition id is 437a9c...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_SNo_prop1)
∀x y, SNo x → SNo y → SNo (x + y) ∧ (∀u ∈ SNoL x, u + y < x + y) ∧ (∀u ∈ SNoR x, x + y < u + y) ∧ (∀u ∈ SNoL y, x + u < x + y) ∧ (∀u ∈ SNoR y, x + y < x + u) ∧ SNoCutP ({w + y|w ∈ SNoL x} ∪ {x + w|w ∈ SNoL y}) ({z + y|z ∈ SNoR x} ∪ {x + z|z ∈ SNoR y})
In Proofgold the corresponding term root is f3ece7... and proposition id is 5a879f...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNo_add_SNo)
∀x y, SNo x → SNo y → SNo (x + y)
In Proofgold the corresponding term root is c4a9a6... and proposition id is a57095...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNo_add_SNo_3)
∀x y z, SNo x → SNo y → SNo z → SNo (x + y + z)
In Proofgold the corresponding term root is f6ef07... and proposition id is f2d78c...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNo_add_SNo_3c)
∀x y z, SNo x → SNo y → SNo z → SNo (x + y + - z)
In Proofgold the corresponding term root is 104eb9... and proposition id is 833be8...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNo_add_SNo_4)
∀x y z w, SNo x → SNo y → SNo z → SNo w → SNo (x + y + z + w)
In Proofgold the corresponding term root is d71f0e... and proposition id is 42037e...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_SNo_Lt1)
∀x y z, SNo x → SNo y → SNo z → x < z → x + y < z + y
In Proofgold the corresponding term root is 92232f... and proposition id is cb0738...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_SNo_Le1)
∀x y z, SNo x → SNo y → SNo z → x ≤ z → x + y ≤ z + y
In Proofgold the corresponding term root is d8f37e... and proposition id is abcf72...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_SNo_Lt2)
∀x y z, SNo x → SNo y → SNo z → y < z → x + y < x + z
In Proofgold the corresponding term root is 6ad464... and proposition id is 6f0941...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_SNo_Le2)
∀x y z, SNo x → SNo y → SNo z → y ≤ z → x + y ≤ x + z
In Proofgold the corresponding term root is 14975e... and proposition id is 3f52ba...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_SNo_Lt3a)
∀x y z w, SNo x → SNo y → SNo z → SNo w → x < z → y ≤ w → x + y < z + w
In Proofgold the corresponding term root is b2215f... and proposition id is 18ff91...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_SNo_Lt3b)
∀x y z w, SNo x → SNo y → SNo z → SNo w → x ≤ z → y < w → x + y < z + w
In Proofgold the corresponding term root is 5b4f92... and proposition id is 8fe0d9...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_SNo_Lt3)
∀x y z w, SNo x → SNo y → SNo z → SNo w → x < z → y < w → x + y < z + w
In Proofgold the corresponding term root is 7141a8... and proposition id is 55bdab...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_SNo_Le3)
∀x y z w, SNo x → SNo y → SNo z → SNo w → x ≤ z → y ≤ w → x + y ≤ z + w
In Proofgold the corresponding term root is 0be637... and proposition id is be66ca...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_SNo_SNoCutP)
∀x y, SNo x → SNo y → SNoCutP ({w + y|w ∈ SNoL x} ∪ {x + w|w ∈ SNoL y}) ({z + y|z ∈ SNoR x} ∪ {x + z|z ∈ SNoR y})
In Proofgold the corresponding term root is e2b7d7... and proposition id is f86b1a...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_SNo_com)
∀x y, SNo x → SNo y → x + y = y + x
In Proofgold the corresponding term root is 548814... and proposition id is f017e5...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_SNo_0L)
∀x, SNo x → 0 + x = x
In Proofgold the corresponding term root is 107598... and proposition id is 446bad...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_SNo_0R)
∀x, SNo x → x + 0 = x
In Proofgold the corresponding term root is d8a186... and proposition id is 1a7fa1...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_SNo_minus_SNo_linv)
∀x, SNo x → - x + x = 0
In Proofgold the corresponding term root is 424719... and proposition id is 44e251...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_SNo_minus_SNo_rinv)
∀x, SNo x → x + - x = 0
In Proofgold the corresponding term root is 9e8c44... and proposition id is b2f9cf...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_SNo_ordinal_SNoCutP)
∀alpha, ordinal alpha → ∀beta, ordinal beta → SNoCutP ({x + beta|x ∈ SNoS_ alpha} ∪ {alpha + x|x ∈ SNoS_ beta}) Empty
In Proofgold the corresponding term root is 2823e4... and proposition id is 507eab...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_SNo_ordinal_eq)
∀alpha, ordinal alpha → ∀beta, ordinal beta → alpha + beta = SNoCut ({x + beta|x ∈ SNoS_ alpha} ∪ {alpha + x|x ∈ SNoS_ beta}) Empty
In Proofgold the corresponding term root is cfe1f5... and proposition id is ee8e7c...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_SNo_ordinal_ordinal)
∀alpha, ordinal alpha → ∀beta, ordinal beta → ordinal (alpha + beta)
In Proofgold the corresponding term root is e9309a... and proposition id is ee9c23...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_SNo_ordinal_SL)
∀alpha, ordinal alpha → ∀beta, ordinal beta → ordsucc alpha + beta = ordsucc (alpha + beta)
In Proofgold the corresponding term root is d55bd8... and proposition id is 61ae57...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_SNo_ordinal_SR)
∀alpha, ordinal alpha → ∀beta, ordinal beta → alpha + ordsucc beta = ordsucc (alpha + beta)
In Proofgold the corresponding term root is e7378a... and proposition id is 0c1a6d...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_SNo_ordinal_InL)
∀alpha, ordinal alpha → ∀beta, ordinal beta → ∀gamma ∈ alpha, gamma + beta ∈ alpha + beta
In Proofgold the corresponding term root is 09a829... and proposition id is 5acda8...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_SNo_ordinal_InR)
∀alpha, ordinal alpha → ∀beta, ordinal beta → ∀gamma ∈ beta, alpha + gamma ∈ alpha + beta
In Proofgold the corresponding term root is d291f4... and proposition id is 0f42b5...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_nat_add_SNo)
∀n m ∈ ω, add_nat n m = n + m
In Proofgold the corresponding term root is 1701fc... and proposition id is 9df659...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_SNo_In_omega)
∀n m ∈ ω, n + m ∈ ω
In Proofgold the corresponding term root is f45cbb... and proposition id is eb6375...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_SNo_1_1_2)
1 + 1 = 2
In Proofgold the corresponding term root is bbc448... and proposition id is 2b8298...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_SNo_SNoL_interpolate)
∀x y, SNo x → SNo y → ∀u ∈ SNoL (x + y), (∃v ∈ SNoL x, u ≤ v + y) ∨ (∃v ∈ SNoL y, u ≤ x + v)
In Proofgold the corresponding term root is 971363... and proposition id is 3cafc1...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_SNo_SNoR_interpolate)
∀x y, SNo x → SNo y → ∀u ∈ SNoR (x + y), (∃v ∈ SNoR x, v + y ≤ u) ∨ (∃v ∈ SNoR y, x + v ≤ u)
In Proofgold the corresponding term root is 928f71... and proposition id is 31fa94...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_SNo_assoc)
∀x y z, SNo x → SNo y → SNo z → x + (y + z) = (x + y) + z
In Proofgold the corresponding term root is 39b331... and proposition id is bc15ad...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_SNo_minus_R2)
∀x y, SNo x → SNo y → (x + y) + - y = x
In Proofgold the corresponding term root is 3e7b33... and proposition id is 91f4ef...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_SNo_minus_R2')
∀x y, SNo x → SNo y → (x + - y) + y = x
In Proofgold the corresponding term root is 4f9468... and proposition id is 43b4d1...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_SNo_minus_L2)
∀x y, SNo x → SNo y → - x + (x + y) = y
In Proofgold the corresponding term root is 2cd65e... and proposition id is 598ee6...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_SNo_minus_L2')
∀x y, SNo x → SNo y → x + (- x + y) = y
In Proofgold the corresponding term root is 632a4d... and proposition id is a8450d...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_SNo_cancel_L)
∀x y z, SNo x → SNo y → SNo z → x + y = x + z → y = z
In Proofgold the corresponding term root is 014096... and proposition id is 781256...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_SNo_cancel_R)
∀x y z, SNo x → SNo y → SNo z → x + y = z + y → x = z
In Proofgold the corresponding term root is 3de43c... and proposition id is e62801...
Proof:
The rest of this subproof is missing.
∎
Theorem. (minus_SNo_0)
- 0 = 0
In Proofgold the corresponding term root is 77bc67... and proposition id is 3fcc57...
Proof:
The rest of this subproof is missing.
∎
Theorem. (minus_add_SNo_distr)
∀x y, SNo x → SNo y → - (x + y) = (- x) + (- y)
In Proofgold the corresponding term root is 57ec9d... and proposition id is da8d5a...
Proof:
The rest of this subproof is missing.
∎
Theorem. (minus_add_SNo_distr_3)
∀x y z, SNo x → SNo y → SNo z → - (x + y + z) = - x + - y + - z
In Proofgold the corresponding term root is b074ac... and proposition id is 94dbf3...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_SNo_Lev_bd)
∀x y, SNo x → SNo y → SNoLev (x + y) ⊆ SNoLev x + SNoLev y
In Proofgold the corresponding term root is 0f257c... and proposition id is 0680ef...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_SNo_SNoS_omega)
∀x y ∈ SNoS_ ω, x + y ∈ SNoS_ ω
In Proofgold the corresponding term root is 931103... and proposition id is 6d16aa...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_SNo_Lt1_cancel)
∀x y z, SNo x → SNo y → SNo z → x + y < z + y → x < z
In Proofgold the corresponding term root is e5c11e... and proposition id is 2ec8ae...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_SNo_Lt2_cancel)
∀x y z, SNo x → SNo y → SNo z → x + y < x + z → y < z
In Proofgold the corresponding term root is 9ecfc8... and proposition id is d4afbd...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_SNo_Le1_cancel)
∀x y z, SNo x → SNo y → SNo z → x + y ≤ z + y → x ≤ z
In Proofgold the corresponding term root is 01a9b8... and proposition id is 8bfac1...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_SNo_assoc_4)
∀x y z w, SNo x → SNo y → SNo z → SNo w → x + y + z + w = (x + y + z) + w
In Proofgold the corresponding term root is b9919a... and proposition id is f2ec8c...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_SNo_com_3_0_1)
∀x y z, SNo x → SNo y → SNo z → x + y + z = y + x + z
In Proofgold the corresponding term root is 53adb5... and proposition id is 3c9cc8...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_SNo_com_3b_1_2)
∀x y z, SNo x → SNo y → SNo z → (x + y) + z = (x + z) + y
In Proofgold the corresponding term root is 2ccf2b... and proposition id is f2dd80...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_SNo_com_4_inner_mid)
∀x y z w, SNo x → SNo y → SNo z → SNo w → (x + y) + (z + w) = (x + z) + (y + w)
In Proofgold the corresponding term root is 635cb7... and proposition id is e25ee6...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_SNo_rotate_3_1)
∀x y z, SNo x → SNo y → SNo z → x + y + z = z + x + y
In Proofgold the corresponding term root is 51654e... and proposition id is 340640...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_SNo_rotate_4_1)
∀x y z w, SNo x → SNo y → SNo z → SNo w → x + y + z + w = w + x + y + z
In Proofgold the corresponding term root is 4bc7aa... and proposition id is ce0a6b...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_SNo_rotate_5_1)
∀x y z w v, SNo x → SNo y → SNo z → SNo w → SNo v → x + y + z + w + v = v + x + y + z + w
In Proofgold the corresponding term root is 4b4b64... and proposition id is 0714aa...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_SNo_rotate_5_2)
∀x y z w v, SNo x → SNo y → SNo z → SNo w → SNo v → x + y + z + w + v = w + v + x + y + z
In Proofgold the corresponding term root is 33b5cc... and proposition id is fec9d0...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_SNo_minus_SNo_prop2)
∀x y, SNo x → SNo y → x + - x + y = y
In Proofgold the corresponding term root is 632a4d... and proposition id is a8450d...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_SNo_minus_SNo_prop3)
∀x y z w, SNo x → SNo y → SNo z → SNo w → (x + y + z) + (- z + w) = x + y + w
In Proofgold the corresponding term root is 038fdd... and proposition id is 0d4f24...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_SNo_minus_SNo_prop5)
∀x y z w, SNo x → SNo y → SNo z → SNo w → (x + y + - z) + (z + w) = x + y + w
In Proofgold the corresponding term root is bdfd3f... and proposition id is eeb587...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_SNo_minus_Lt1)
∀x y z, SNo x → SNo y → SNo z → x + - y < z → x < z + y
In Proofgold the corresponding term root is 643876... and proposition id is 4033dc...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_SNo_minus_Lt2)
∀x y z, SNo x → SNo y → SNo z → z < x + - y → z + y < x
In Proofgold the corresponding term root is 76c543... and proposition id is 362145...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_SNo_minus_Lt1b)
∀x y z, SNo x → SNo y → SNo z → x < z + y → x + - y < z
In Proofgold the corresponding term root is 5012b1... and proposition id is 0b5cca...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_SNo_minus_Lt2b)
∀x y z, SNo x → SNo y → SNo z → z + y < x → z < x + - y
In Proofgold the corresponding term root is 24d1a6... and proposition id is 684192...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_SNo_minus_Lt1b3)
∀x y z w, SNo x → SNo y → SNo z → SNo w → x + y < w + z → x + y + - z < w
In Proofgold the corresponding term root is 218e4f... and proposition id is 8ea4a7...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_SNo_minus_Lt2b3)
∀x y z w, SNo x → SNo y → SNo z → SNo w → w + z < x + y → w < x + y + - z
In Proofgold the corresponding term root is d9462b... and proposition id is 4bf575...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_SNo_minus_Lt_lem)
∀x y z u v w, SNo x → SNo y → SNo z → SNo u → SNo v → SNo w → x + y + w < u + v + z → x + y + - z < u + v + - w
In Proofgold the corresponding term root is da9f29... and proposition id is d29d2a...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_SNo_minus_Le2)
∀x y z, SNo x → SNo y → SNo z → z ≤ x + - y → z + y ≤ x
In Proofgold the corresponding term root is 85af84... and proposition id is 86b40b...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_SNo_minus_Le2b)
∀x y z, SNo x → SNo y → SNo z → z + y ≤ x → z ≤ x + - y
In Proofgold the corresponding term root is 688ef9... and proposition id is d815dd...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_SNo_Lt_subprop2)
∀x y z w u v, SNo x → SNo y → SNo z → SNo w → SNo u → SNo v → x + u < z + v → y + v < w + u → x + y < z + w
In Proofgold the corresponding term root is 848909... and proposition id is 7f1b50...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_SNo_Lt_subprop3a)
∀x y z w u a, SNo x → SNo y → SNo z → SNo w → SNo u → SNo a → x + z < w + a → y + a < u → x + y + z < w + u
In Proofgold the corresponding term root is 43322d... and proposition id is cde0c9...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_SNo_Lt_subprop3b)
∀x y w u v a, SNo x → SNo y → SNo w → SNo u → SNo v → SNo a → x + a < w + v → y < a + u → x + y < w + u + v
In Proofgold the corresponding term root is 9d5c37... and proposition id is b197b1...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_SNo_Lt_subprop3c)
∀x y z w u a b c, SNo x → SNo y → SNo z → SNo w → SNo u → SNo a → SNo b → SNo c → x + a < b + c → y + c < u → b + z < w + a → x + y + z < w + u
In Proofgold the corresponding term root is 184614... and proposition id is f8cf2b...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_SNo_Lt_subprop3d)
∀x y w u v a b c, SNo x → SNo y → SNo w → SNo u → SNo v → SNo a → SNo b → SNo c → x + a < b + v → y < c + u → b + c < w + a → x + y < w + u + v
In Proofgold the corresponding term root is a447c6... and proposition id is a1520a...
Proof:
The rest of this subproof is missing.
∎
Theorem. (ordinal_ordsucc_SNo_eq)
∀alpha, ordinal alpha → ordsucc alpha = 1 + alpha
In Proofgold the corresponding term root is f0596d... and proposition id is 36bf6d...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_SNo_3a_2b)
∀x y z w u, SNo x → SNo y → SNo z → SNo w → SNo u → (x + y + z) + (w + u) = (u + y + z) + (w + x)
In Proofgold the corresponding term root is d5c6f9... and proposition id is f24a27...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_SNo_1_ordsucc)
∀n ∈ ω, n + 1 = ordsucc n
In Proofgold the corresponding term root is ec6cad... and proposition id is 85f670...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_SNo_eps_Lt)
∀x, SNo x → ∀n ∈ ω, x < x + eps_ n
In Proofgold the corresponding term root is dfe40b... and proposition id is d492a4...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_SNo_eps_Lt')
∀x y, SNo x → SNo y → ∀n ∈ ω, x < y → x < y + eps_ n
In Proofgold the corresponding term root is 281ca0... and proposition id is 943874...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNoLt_minus_pos)
∀x y, SNo x → SNo y → x < y → 0 < y + - x
In Proofgold the corresponding term root is fff649... and proposition id is 2c799f...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_SNo_omega_In_cases)
∀m, ∀n ∈ ω, ∀k, nat_p k → m ∈ n + k → m ∈ n ∨ m + - n ∈ k
In Proofgold the corresponding term root is 3eee93... and proposition id is 584cf6...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_SNo_Lt4)
∀x y z w u v, SNo x → SNo y → SNo z → SNo w → SNo u → SNo v → x < w → y < u → z < v → x + y + z < w + u + v
In Proofgold the corresponding term root is 7e45aa... and proposition id is c85ce6...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_SNo_3_3_3_Lt1)
∀x y z w u, SNo x → SNo y → SNo z → SNo w → SNo u → x + y < z + w → x + y + u < z + w + u
In Proofgold the corresponding term root is 5889f8... and proposition id is fa11ed...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_SNo_3_2_3_Lt1)
∀x y z w u, SNo x → SNo y → SNo z → SNo w → SNo u → y + x < z + w → x + u + y < z + w + u
In Proofgold the corresponding term root is c05b73... and proposition id is e4013b...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_SNo_minus_Lt12b3)
∀x y z w u v, SNo x → SNo y → SNo z → SNo w → SNo u → SNo v → x + y + v < w + u + z → x + y + - z < w + u + - v
In Proofgold the corresponding term root is da9f29... and proposition id is d29d2a...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_SNo_minus_Le1b)
∀x y z, SNo x → SNo y → SNo z → x ≤ z + y → x + - y ≤ z
In Proofgold the corresponding term root is c8c038... and proposition id is 6a5e5e...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_SNo_minus_Le1b3)
∀x y z w, SNo x → SNo y → SNo z → SNo w → x + y ≤ w + z → x + y + - z ≤ w
In Proofgold the corresponding term root is 11c341... and proposition id is 6a25dd...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_SNo_minus_Le12b3)
∀x y z w u v, SNo x → SNo y → SNo z → SNo w → SNo u → SNo v → x + y + v ≤ w + u + z → x + y + - z ≤ w + u + - v
In Proofgold the corresponding term root is 055445... and proposition id is d88d85...
Proof:
The rest of this subproof is missing.
∎
End of Section SurrealAdd
Beginning of Section SurrealAbs
Notation. We use - as a prefix operator with priority 358 corresponding to applying term minus_SNo.
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_SNo.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_SNo.
Definition. We define abs_SNo to be λx ⇒ if 0 ≤ x then x else - x of type set → set.
In Proofgold the corresponding term root is 34f6dc... and object id is 2990e5...
Theorem. (nonneg_abs_SNo)
∀x, 0 ≤ x → abs_SNo x = x
In Proofgold the corresponding term root is 330ed1... and proposition id is 1ed323...
Proof:
The rest of this subproof is missing.
∎
Theorem. (not_nonneg_abs_SNo)
∀x, ¬ (0 ≤ x) → abs_SNo x = - x
In Proofgold the corresponding term root is bc755d... and proposition id is b29f70...
Proof:
The rest of this subproof is missing.
∎
Theorem. (pos_abs_SNo)
∀x, 0 < x → abs_SNo x = x
In Proofgold the corresponding term root is 32e626... and proposition id is 88568f...
Proof:
The rest of this subproof is missing.
∎
Theorem. (neg_abs_SNo)
∀x, SNo x → x < 0 → abs_SNo x = - x
In Proofgold the corresponding term root is d1245a... and proposition id is f858cb...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNo_abs_SNo)
∀x, SNo x → SNo (abs_SNo x)
In Proofgold the corresponding term root is 7266a7... and proposition id is 1c9c46...
Proof:
The rest of this subproof is missing.
∎
Theorem. (abs_SNo_minus)
∀x, SNo x → abs_SNo (- x) = abs_SNo x
In Proofgold the corresponding term root is 977106... and proposition id is 2c0d4c...
Proof:
The rest of this subproof is missing.
∎
Theorem. (abs_SNo_dist_swap)
∀x y, SNo x → SNo y → abs_SNo (x + - y) = abs_SNo (y + - x)
In Proofgold the corresponding term root is 73d709... and proposition id is fd708c...
Proof:
The rest of this subproof is missing.
∎
End of Section SurrealAbs
Beginning of Section SurrealMul
Notation. We use - as a prefix operator with priority 358 corresponding to applying term minus_SNo.
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_SNo.
Definition. We define mul_SNo to be SNo_rec2 (λx y m ⇒ SNoCut ({m (w 0) y + m x (w 1) + - m (w 0) (w 1)|w ∈ SNoL x ⨯ SNoL y} ∪ {m (z 0) y + m x (z 1) + - m (z 0) (z 1)|z ∈ SNoR x ⨯ SNoR y}) ({m (w 0) y + m x (w 1) + - m (w 0) (w 1)|w ∈ SNoL x ⨯ SNoR y} ∪ {m (z 0) y + m x (z 1) + - m (z 0) (z 1)|z ∈ SNoR x ⨯ SNoL y})) of type set → set → set.
In Proofgold the corresponding term root is 48d054... and object id is b14730...
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_SNo.
Theorem. (mul_SNo_eq)
∀x, SNo x → ∀y, SNo y → x * y = SNoCut ({(w 0) * y + x * (w 1) + - (w 0) * (w 1)|w ∈ SNoL x ⨯ SNoL y} ∪ {(z 0) * y + x * (z 1) + - (z 0) * (z 1)|z ∈ SNoR x ⨯ SNoR y}) ({(w 0) * y + x * (w 1) + - (w 0) * (w 1)|w ∈ SNoL x ⨯ SNoR y} ∪ {(z 0) * y + x * (z 1) + - (z 0) * (z 1)|z ∈ SNoR x ⨯ SNoL y})
In Proofgold the corresponding term root is 726bd9... and proposition id is 3525c5...
Proof:
The rest of this subproof is missing.
∎
Theorem. (mul_SNo_eq_2)
∀x y, SNo x → SNo y → ∀p : prop, (∀L R, (∀u, u ∈ L → (∀q : prop, (∀w0 ∈ SNoL x, ∀w1 ∈ SNoL y, u = w0 * y + x * w1 + - w0 * w1 → q) → (∀z0 ∈ SNoR x, ∀z1 ∈ SNoR y, u = z0 * y + x * z1 + - z0 * z1 → q) → q)) → (∀w0 ∈ SNoL x, ∀w1 ∈ SNoL y, w0 * y + x * w1 + - w0 * w1 ∈ L) → (∀z0 ∈ SNoR x, ∀z1 ∈ SNoR y, z0 * y + x * z1 + - z0 * z1 ∈ L) → (∀u, u ∈ R → (∀q : prop, (∀w0 ∈ SNoL x, ∀z1 ∈ SNoR y, u = w0 * y + x * z1 + - w0 * z1 → q) → (∀z0 ∈ SNoR x, ∀w1 ∈ SNoL y, u = z0 * y + x * w1 + - z0 * w1 → q) → q)) → (∀w0 ∈ SNoL x, ∀z1 ∈ SNoR y, w0 * y + x * z1 + - w0 * z1 ∈ R) → (∀z0 ∈ SNoR x, ∀w1 ∈ SNoL y, z0 * y + x * w1 + - z0 * w1 ∈ R) → x * y = SNoCut L R → p) → p
In Proofgold the corresponding term root is 97f72a... and proposition id is d69cc6...
Proof:
The rest of this subproof is missing.
∎
Theorem. (mul_SNo_prop_1)
∀x, SNo x → ∀y, SNo y → ∀p : prop, (SNo (x * y) → (∀u ∈ SNoL x, ∀v ∈ SNoL y, u * y + x * v < x * y + u * v) → (∀u ∈ SNoR x, ∀v ∈ SNoR y, u * y + x * v < x * y + u * v) → (∀u ∈ SNoL x, ∀v ∈ SNoR y, x * y + u * v < u * y + x * v) → (∀u ∈ SNoR x, ∀v ∈ SNoL y, x * y + u * v < u * y + x * v) → p) → p
In Proofgold the corresponding term root is 3018e4... and proposition id is 5e4d16...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNo_mul_SNo)
∀x y, SNo x → SNo y → SNo (x * y)
In Proofgold the corresponding term root is 7fa2e2... and proposition id is 9fd082...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNo_mul_SNo_lem)
∀x y u v, SNo x → SNo y → SNo u → SNo v → SNo (u * y + x * v + - (u * v))
In Proofgold the corresponding term root is 632e95... and proposition id is 3a9abb...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNo_mul_SNo_3)
∀x y z, SNo x → SNo y → SNo z → SNo (x * y * z)
In Proofgold the corresponding term root is 4fc87d... and proposition id is 8f5450...
Proof:
The rest of this subproof is missing.
∎
Theorem. (mul_SNo_eq_3)
∀x y, SNo x → SNo y → ∀p : prop, (∀L R, SNoCutP L R → (∀u, u ∈ L → (∀q : prop, (∀w0 ∈ SNoL x, ∀w1 ∈ SNoL y, u = w0 * y + x * w1 + - w0 * w1 → q) → (∀z0 ∈ SNoR x, ∀z1 ∈ SNoR y, u = z0 * y + x * z1 + - z0 * z1 → q) → q)) → (∀w0 ∈ SNoL x, ∀w1 ∈ SNoL y, w0 * y + x * w1 + - w0 * w1 ∈ L) → (∀z0 ∈ SNoR x, ∀z1 ∈ SNoR y, z0 * y + x * z1 + - z0 * z1 ∈ L) → (∀u, u ∈ R → (∀q : prop, (∀w0 ∈ SNoL x, ∀z1 ∈ SNoR y, u = w0 * y + x * z1 + - w0 * z1 → q) → (∀z0 ∈ SNoR x, ∀w1 ∈ SNoL y, u = z0 * y + x * w1 + - z0 * w1 → q) → q)) → (∀w0 ∈ SNoL x, ∀z1 ∈ SNoR y, w0 * y + x * z1 + - w0 * z1 ∈ R) → (∀z0 ∈ SNoR x, ∀w1 ∈ SNoL y, z0 * y + x * w1 + - z0 * w1 ∈ R) → x * y = SNoCut L R → p) → p
In Proofgold the corresponding term root is 517939... and proposition id is 720ba0...
Proof:
The rest of this subproof is missing.
∎
Theorem. (mul_SNo_Lt)
∀x y u v, SNo x → SNo y → SNo u → SNo v → u < x → v < y → u * y + x * v < x * y + u * v
In Proofgold the corresponding term root is 428b35... and proposition id is c23201...
Proof:
The rest of this subproof is missing.
∎
Theorem. (mul_SNo_Le)
∀x y u v, SNo x → SNo y → SNo u → SNo v → u ≤ x → v ≤ y → u * y + x * v ≤ x * y + u * v
In Proofgold the corresponding term root is d9a12c... and proposition id is acb620...
Proof:
The rest of this subproof is missing.
∎
Theorem. (mul_SNo_SNoL_interpolate)
∀x y, SNo x → SNo y → ∀u ∈ SNoL (x * y), (∃v ∈ SNoL x, ∃w ∈ SNoL y, u + v * w ≤ v * y + x * w) ∨ (∃v ∈ SNoR x, ∃w ∈ SNoR y, u + v * w ≤ v * y + x * w)
In Proofgold the corresponding term root is 7d1b52... and proposition id is f460fc...
Proof:
The rest of this subproof is missing.
∎
Theorem. (mul_SNo_SNoL_interpolate_impred)
∀x y, SNo x → SNo y → ∀u ∈ SNoL (x * y), ∀p : prop, (∀v ∈ SNoL x, ∀w ∈ SNoL y, u + v * w ≤ v * y + x * w → p) → (∀v ∈ SNoR x, ∀w ∈ SNoR y, u + v * w ≤ v * y + x * w → p) → p
In Proofgold the corresponding term root is 395b74... and proposition id is ad6f96...
Proof:
The rest of this subproof is missing.
∎
Theorem. (mul_SNo_SNoR_interpolate)
∀x y, SNo x → SNo y → ∀u ∈ SNoR (x * y), (∃v ∈ SNoL x, ∃w ∈ SNoR y, v * y + x * w ≤ u + v * w) ∨ (∃v ∈ SNoR x, ∃w ∈ SNoL y, v * y + x * w ≤ u + v * w)
In Proofgold the corresponding term root is e15949... and proposition id is b67a35...
Proof:
The rest of this subproof is missing.
∎
Theorem. (mul_SNo_SNoR_interpolate_impred)
∀x y, SNo x → SNo y → ∀u ∈ SNoR (x * y), ∀p : prop, (∀v ∈ SNoL x, ∀w ∈ SNoR y, v * y + x * w ≤ u + v * w → p) → (∀v ∈ SNoR x, ∀w ∈ SNoL y, v * y + x * w ≤ u + v * w → p) → p
In Proofgold the corresponding term root is e5cae5... and proposition id is d1fff1...
Proof:
The rest of this subproof is missing.
∎
Theorem. (mul_SNo_Subq_lem)
∀x y X Y Z W, ∀U U', (∀u, u ∈ U → (∀q : prop, (∀w0 ∈ X, ∀w1 ∈ Y, u = w0 * y + x * w1 + - w0 * w1 → q) → (∀z0 ∈ Z, ∀z1 ∈ W, u = z0 * y + x * z1 + - z0 * z1 → q) → q)) → (∀w0 ∈ X, ∀w1 ∈ Y, w0 * y + x * w1 + - w0 * w1 ∈ U') → (∀w0 ∈ Z, ∀w1 ∈ W, w0 * y + x * w1 + - w0 * w1 ∈ U') → U ⊆ U'
In Proofgold the corresponding term root is cc8343... and proposition id is e8bfa2...
Proof:
The rest of this subproof is missing.
∎
Theorem. (mul_SNo_zeroR)
∀x, SNo x → x * 0 = 0
In Proofgold the corresponding term root is d279f4... and proposition id is a4f4a6...
Proof:
The rest of this subproof is missing.
∎
Theorem. (mul_SNo_oneR)
∀x, SNo x → x * 1 = x
In Proofgold the corresponding term root is 1d2f1d... and proposition id is 5477a4...
Proof:
The rest of this subproof is missing.
∎
Theorem. (mul_SNo_com)
∀x y, SNo x → SNo y → x * y = y * x
In Proofgold the corresponding term root is 21f7d8... and proposition id is 83d5fa...
Proof:
The rest of this subproof is missing.
∎
Theorem. (mul_SNo_minus_distrL)
∀x y, SNo x → SNo y → (- x) * y = - x * y
In Proofgold the corresponding term root is 9beaa0... and proposition id is 3d0cd1...
Proof:
The rest of this subproof is missing.
∎
Theorem. (mul_SNo_minus_distrR)
∀x y, SNo x → SNo y → x * (- y) = - (x * y)
In Proofgold the corresponding term root is ab775a... and proposition id is 2e7475...
Proof:
The rest of this subproof is missing.
∎
Theorem. (mul_SNo_distrR)
∀x y z, SNo x → SNo y → SNo z → (x + y) * z = x * z + y * z
In Proofgold the corresponding term root is 6a23e5... and proposition id is 5f0382...
Proof:
The rest of this subproof is missing.
∎
Theorem. (mul_SNo_distrL)
∀x y z, SNo x → SNo y → SNo z → x * (y + z) = x * y + x * z
In Proofgold the corresponding term root is 4e7eba... and proposition id is 2a4834...
Proof:
The rest of this subproof is missing.
∎
Beginning of Section mul_SNo_assoc_lems
Variable M : set → set → set
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term M.
Hypothesis SNo_M : ∀x y, SNo x → SNo y → SNo (x * y)
Hypothesis DL : ∀x y z, SNo x → SNo y → SNo z → x * (y + z) = x * y + x * z
Hypothesis DR : ∀x y z, SNo x → SNo y → SNo z → (x + y) * z = x * z + y * z
Hypothesis IL : ∀x y, SNo x → SNo y → ∀u ∈ SNoL (x * y), ∀p : prop, (∀v ∈ SNoL x, ∀w ∈ SNoL y, u + v * w ≤ v * y + x * w → p) → (∀v ∈ SNoR x, ∀w ∈ SNoR y, u + v * w ≤ v * y + x * w → p) → p
Hypothesis IR : ∀x y, SNo x → SNo y → ∀u ∈ SNoR (x * y), ∀p : prop, (∀v ∈ SNoL x, ∀w ∈ SNoR y, v * y + x * w ≤ u + v * w → p) → (∀v ∈ SNoR x, ∀w ∈ SNoL y, v * y + x * w ≤ u + v * w → p) → p
Hypothesis M_Lt : ∀x y u v, SNo x → SNo y → SNo u → SNo v → u < x → v < y → u * y + x * v < x * y + u * v
Hypothesis M_Le : ∀x y u v, SNo x → SNo y → SNo u → SNo v → u ≤ x → v ≤ y → u * y + x * v ≤ x * y + u * v
Theorem. (mul_SNo_assoc_lem1)
∀x y z, SNo x → SNo y → SNo z → (∀u ∈ SNoS_ (SNoLev x), u * (y * z) = (u * y) * z) → (∀v ∈ SNoS_ (SNoLev y), x * (v * z) = (x * v) * z) → (∀w ∈ SNoS_ (SNoLev z), x * (y * w) = (x * y) * w) → (∀u ∈ SNoS_ (SNoLev x), ∀v ∈ SNoS_ (SNoLev y), u * (v * z) = (u * v) * z) → (∀u ∈ SNoS_ (SNoLev x), ∀w ∈ SNoS_ (SNoLev z), u * (y * w) = (u * y) * w) → (∀v ∈ SNoS_ (SNoLev y), ∀w ∈ SNoS_ (SNoLev z), x * (v * w) = (x * v) * w) → (∀u ∈ SNoS_ (SNoLev x), ∀v ∈ SNoS_ (SNoLev y), ∀w ∈ SNoS_ (SNoLev z), u * (v * w) = (u * v) * w) → ∀L, (∀u ∈ L, ∀q : prop, (∀v ∈ SNoL x, ∀w ∈ SNoL (y * z), u = v * (y * z) + x * w + - v * w → q) → (∀v ∈ SNoR x, ∀w ∈ SNoR (y * z), u = v * (y * z) + x * w + - v * w → q) → q) → ∀u ∈ L, u < (x * y) * z
In Proofgold the corresponding term root is fca036... and proposition id is b70da8...
Proof:
The rest of this subproof is missing.
∎
Theorem. (mul_SNo_assoc_lem2)
∀x y z, SNo x → SNo y → SNo z → (∀u ∈ SNoS_ (SNoLev x), u * (y * z) = (u * y) * z) → (∀v ∈ SNoS_ (SNoLev y), x * (v * z) = (x * v) * z) → (∀w ∈ SNoS_ (SNoLev z), x * (y * w) = (x * y) * w) → (∀u ∈ SNoS_ (SNoLev x), ∀v ∈ SNoS_ (SNoLev y), u * (v * z) = (u * v) * z) → (∀u ∈ SNoS_ (SNoLev x), ∀w ∈ SNoS_ (SNoLev z), u * (y * w) = (u * y) * w) → (∀v ∈ SNoS_ (SNoLev y), ∀w ∈ SNoS_ (SNoLev z), x * (v * w) = (x * v) * w) → (∀u ∈ SNoS_ (SNoLev x), ∀v ∈ SNoS_ (SNoLev y), ∀w ∈ SNoS_ (SNoLev z), u * (v * w) = (u * v) * w) → ∀R, (∀u ∈ R, ∀q : prop, (∀v ∈ SNoL x, ∀w ∈ SNoR (y * z), u = v * (y * z) + x * w + - v * w → q) → (∀v ∈ SNoR x, ∀w ∈ SNoL (y * z), u = v * (y * z) + x * w + - v * w → q) → q) → ∀u ∈ R, (x * y) * z < u
In Proofgold the corresponding term root is 749b30... and proposition id is f249d3...
Proof:
The rest of this subproof is missing.
∎
End of Section mul_SNo_assoc_lems
Theorem. (mul_SNo_assoc)
∀x y z, SNo x → SNo y → SNo z → x * (y * z) = (x * y) * z
In Proofgold the corresponding term root is 71a504... and proposition id is d322c1...
Proof:
The rest of this subproof is missing.
∎
Theorem. (mul_nat_mul_SNo)
∀n m ∈ ω, mul_nat n m = n * m
In Proofgold the corresponding term root is da39d3... and proposition id is eb6335...
Proof:
The rest of this subproof is missing.
∎
Theorem. (mul_SNo_In_omega)
∀n m ∈ ω, n * m ∈ ω
In Proofgold the corresponding term root is 7b6023... and proposition id is 8435a4...
Proof:
The rest of this subproof is missing.
∎
Theorem. (mul_SNo_zeroL)
∀x, SNo x → 0 * x = 0
In Proofgold the corresponding term root is ddc2f5... and proposition id is aaf0da...
Proof:
The rest of this subproof is missing.
∎
Theorem. (mul_SNo_oneL)
∀x, SNo x → 1 * x = x
In Proofgold the corresponding term root is 6f8b10... and proposition id is 1b25c9...
Proof:
The rest of this subproof is missing.
∎
Theorem. (mul_SNo_rotate_3_1)
∀x y z, SNo x → SNo y → SNo z → x * y * z = z * x * y
In Proofgold the corresponding term root is 920309... and proposition id is 15ee6c...
Proof:
The rest of this subproof is missing.
∎
Theorem. (pos_mul_SNo_Lt)
∀x y z, SNo x → 0 < x → SNo y → SNo z → y < z → x * y < x * z
In Proofgold the corresponding term root is 674213... and proposition id is 0e949e...
Proof:
The rest of this subproof is missing.
∎
Theorem. (nonneg_mul_SNo_Le)
∀x y z, SNo x → 0 ≤ x → SNo y → SNo z → y ≤ z → x * y ≤ x * z
In Proofgold the corresponding term root is b8880b... and proposition id is ae94b7...
Proof:
The rest of this subproof is missing.
∎
Theorem. (neg_mul_SNo_Lt)
∀x y z, SNo x → x < 0 → SNo y → SNo z → z < y → x * y < x * z
In Proofgold the corresponding term root is 713502... and proposition id is 63326a...
Proof:
The rest of this subproof is missing.
∎
Theorem. (pos_mul_SNo_Lt')
∀x y z, SNo x → SNo y → SNo z → 0 < z → x < y → x * z < y * z
In Proofgold the corresponding term root is a08fe5... and proposition id is 436be0...
Proof:
The rest of this subproof is missing.
∎
Theorem. (mul_SNo_Lt1_pos_Lt)
∀x y, SNo x → SNo y → x < 1 → 0 < y → x * y < y
In Proofgold the corresponding term root is c5e733... and proposition id is 9c395c...
Proof:
The rest of this subproof is missing.
∎
Theorem. (nonneg_mul_SNo_Le')
∀x y z, SNo x → SNo y → SNo z → 0 ≤ z → x ≤ y → x * z ≤ y * z
In Proofgold the corresponding term root is 1577b1... and proposition id is e85848...
Proof:
The rest of this subproof is missing.
∎
Theorem. (mul_SNo_Le1_nonneg_Le)
∀x y, SNo x → SNo y → x ≤ 1 → 0 ≤ y → x * y ≤ y
In Proofgold the corresponding term root is 26eb53... and proposition id is 655b0c...
Proof:
The rest of this subproof is missing.
∎
Theorem. (pos_mul_SNo_Lt2)
∀x y z w, SNo x → SNo y → SNo z → SNo w → 0 < x → 0 < y → x < z → y < w → x * y < z * w
In Proofgold the corresponding term root is f53e1e... and proposition id is d23a11...
Proof:
The rest of this subproof is missing.
∎
Theorem. (nonneg_mul_SNo_Le2)
∀x y z w, SNo x → SNo y → SNo z → SNo w → 0 ≤ x → 0 ≤ y → x ≤ z → y ≤ w → x * y ≤ z * w
In Proofgold the corresponding term root is 37f3f7... and proposition id is e53b91...
Proof:
The rest of this subproof is missing.
∎
Theorem. (mul_SNo_pos_pos)
∀x y, SNo x → SNo y → 0 < x → 0 < y → 0 < x * y
In Proofgold the corresponding term root is fda250... and proposition id is d984dc...
Proof:
The rest of this subproof is missing.
∎
Theorem. (mul_SNo_pos_neg)
∀x y, SNo x → SNo y → 0 < x → y < 0 → x * y < 0
In Proofgold the corresponding term root is 4dec1e... and proposition id is 697e77...
Proof:
The rest of this subproof is missing.
∎
Theorem. (mul_SNo_neg_pos)
∀x y, SNo x → SNo y → x < 0 → 0 < y → x * y < 0
In Proofgold the corresponding term root is 6f7ff5... and proposition id is 6e7adc...
Proof:
The rest of this subproof is missing.
∎
Theorem. (mul_SNo_neg_neg)
∀x y, SNo x → SNo y → x < 0 → y < 0 → 0 < x * y
In Proofgold the corresponding term root is 86d604... and proposition id is 03517a...
Proof:
The rest of this subproof is missing.
∎
Theorem. (mul_SNo_nonneg_nonneg)
∀x y, SNo x → SNo y → 0 ≤ x → 0 ≤ y → 0 ≤ x * y
In Proofgold the corresponding term root is 53d610... and proposition id is fc765b...
Proof:
The rest of this subproof is missing.
∎
Theorem. (mul_SNo_nonpos_pos)
∀x y, SNo x → SNo y → x ≤ 0 → 0 < y → x * y ≤ 0
In Proofgold the corresponding term root is c597ec... and proposition id is fcd532...
Proof:
The rest of this subproof is missing.
∎
Theorem. (mul_SNo_nonpos_neg)
∀x y, SNo x → SNo y → x ≤ 0 → y < 0 → 0 ≤ x * y
In Proofgold the corresponding term root is 54e1f5... and proposition id is 906128...
Proof:
The rest of this subproof is missing.
∎
Theorem. (nonpos_mul_SNo_Le)
∀x y z, SNo x → x ≤ 0 → SNo y → SNo z → z ≤ y → x * y ≤ x * z
In Proofgold the corresponding term root is ebad85... and proposition id is c0938c...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNo_zero_or_sqr_pos)
∀x, SNo x → x = 0 ∨ 0 < x * x
In Proofgold the corresponding term root is 4929cf... and proposition id is 489689...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNo_pos_sqr_uniq)
∀x y, SNo x → SNo y → 0 < x → 0 < y → x * x = y * y → x = y
In Proofgold the corresponding term root is f41831... and proposition id is 31cc9e...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNo_nonneg_sqr_uniq)
∀x y, SNo x → SNo y → 0 ≤ x → 0 ≤ y → x * x = y * y → x = y
In Proofgold the corresponding term root is 68d62c... and proposition id is d45b76...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNo_foil)
∀x y z w, SNo x → SNo y → SNo z → SNo w → (x + y) * (z + w) = x * z + x * w + y * z + y * w
In Proofgold the corresponding term root is d65f34... and proposition id is ab6bae...
Proof:
The rest of this subproof is missing.
∎
Theorem. (mul_SNo_minus_minus)
∀x y, SNo x → SNo y → (- x) * (- y) = x * y
In Proofgold the corresponding term root is cc94f8... and proposition id is 2674e5...
Proof:
The rest of this subproof is missing.
∎
Theorem. (mul_SNo_com_3_0_1)
∀x y z, SNo x → SNo y → SNo z → x * y * z = y * x * z
In Proofgold the corresponding term root is 00a644... and proposition id is cda6e7...
Proof:
The rest of this subproof is missing.
∎
Theorem. (mul_SNo_com_3b_1_2)
∀x y z, SNo x → SNo y → SNo z → (x * y) * z = (x * z) * y
In Proofgold the corresponding term root is 91c3e7... and proposition id is e48f00...
Proof:
The rest of this subproof is missing.
∎
Theorem. (mul_SNo_com_4_inner_mid)
∀x y z w, SNo x → SNo y → SNo z → SNo w → (x * y) * (z * w) = (x * z) * (y * w)
In Proofgold the corresponding term root is 0bafcb... and proposition id is e23f15...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNo_foil_mm)
∀x y z w, SNo x → SNo y → SNo z → SNo w → (x + - y) * (z + - w) = x * z + - x * w + - y * z + y * w
In Proofgold the corresponding term root is c5dda6... and proposition id is 6b5f54...
Proof:
The rest of this subproof is missing.
∎
Theorem. (mul_SNo_nonzero_cancel)
∀x y z, SNo x → x ≠ 0 → SNo y → SNo z → x * y = x * z → y = z
In Proofgold the corresponding term root is 67c26c... and proposition id is 035c5c...
Proof:
The rest of this subproof is missing.
∎
Theorem. (mul_SNoCutP_lem)
∀Lx Rx Ly Ry x y, SNoCutP Lx Rx → SNoCutP Ly Ry → x = SNoCut Lx Rx → y = SNoCut Ly Ry → SNoCutP ({w 0 * y + x * w 1 + - w 0 * w 1|w ∈ Lx ⨯ Ly} ∪ {z 0 * y + x * z 1 + - z 0 * z 1|z ∈ Rx ⨯ Ry}) ({w 0 * y + x * w 1 + - w 0 * w 1|w ∈ Lx ⨯ Ry} ∪ {z 0 * y + x * z 1 + - z 0 * z 1|z ∈ Rx ⨯ Ly}) ∧ x * y = SNoCut ({w 0 * y + x * w 1 + - w 0 * w 1|w ∈ Lx ⨯ Ly} ∪ {z 0 * y + x * z 1 + - z 0 * z 1|z ∈ Rx ⨯ Ry}) ({w 0 * y + x * w 1 + - w 0 * w 1|w ∈ Lx ⨯ Ry} ∪ {z 0 * y + x * z 1 + - z 0 * z 1|z ∈ Rx ⨯ Ly}) ∧ ∀q : prop, (∀LxLy' RxRy' LxRy' RxLy', (∀u ∈ LxLy', ∀p : prop, (∀w ∈ Lx, ∀w' ∈ Ly, SNo w → SNo w' → w < x → w' < y → u = w * y + x * w' + - w * w' → p) → p) → (∀w ∈ Lx, ∀w' ∈ Ly, w * y + x * w' + - w * w' ∈ LxLy') → (∀u ∈ RxRy', ∀p : prop, (∀z ∈ Rx, ∀z' ∈ Ry, SNo z → SNo z' → x < z → y < z' → u = z * y + x * z' + - z * z' → p) → p) → (∀z ∈ Rx, ∀z' ∈ Ry, z * y + x * z' + - z * z' ∈ RxRy') → (∀u ∈ LxRy', ∀p : prop, (∀w ∈ Lx, ∀z ∈ Ry, SNo w → SNo z → w < x → y < z → u = w * y + x * z + - w * z → p) → p) → (∀w ∈ Lx, ∀z ∈ Ry, w * y + x * z + - w * z ∈ LxRy') → (∀u ∈ RxLy', ∀p : prop, (∀z ∈ Rx, ∀w ∈ Ly, SNo z → SNo w → x < z → w < y → u = z * y + x * w + - z * w → p) → p) → (∀z ∈ Rx, ∀w ∈ Ly, z * y + x * w + - z * w ∈ RxLy') → SNoCutP (LxLy' ∪ RxRy') (LxRy' ∪ RxLy') → x * y = SNoCut (LxLy' ∪ RxRy') (LxRy' ∪ RxLy') → q) → q
In Proofgold the corresponding term root is 912d2d... and proposition id is 4f23eb...
Proof:
The rest of this subproof is missing.
∎
Theorem. (mul_SNoCut_abs)
∀Lx Rx Ly Ry x y, SNoCutP Lx Rx → SNoCutP Ly Ry → x = SNoCut Lx Rx → y = SNoCut Ly Ry → ∀q : prop, (∀LxLy' RxRy' LxRy' RxLy', (∀u ∈ LxLy', ∀p : prop, (∀w ∈ Lx, ∀w' ∈ Ly, SNo w → SNo w' → w < x → w' < y → u = w * y + x * w' + - w * w' → p) → p) → (∀w ∈ Lx, ∀w' ∈ Ly, w * y + x * w' + - w * w' ∈ LxLy') → (∀u ∈ RxRy', ∀p : prop, (∀z ∈ Rx, ∀z' ∈ Ry, SNo z → SNo z' → x < z → y < z' → u = z * y + x * z' + - z * z' → p) → p) → (∀z ∈ Rx, ∀z' ∈ Ry, z * y + x * z' + - z * z' ∈ RxRy') → (∀u ∈ LxRy', ∀p : prop, (∀w ∈ Lx, ∀z ∈ Ry, SNo w → SNo z → w < x → y < z → u = w * y + x * z + - w * z → p) → p) → (∀w ∈ Lx, ∀z ∈ Ry, w * y + x * z + - w * z ∈ LxRy') → (∀u ∈ RxLy', ∀p : prop, (∀z ∈ Rx, ∀w ∈ Ly, SNo z → SNo w → x < z → w < y → u = z * y + x * w + - z * w → p) → p) → (∀z ∈ Rx, ∀w ∈ Ly, z * y + x * w + - z * w ∈ RxLy') → SNoCutP (LxLy' ∪ RxRy') (LxRy' ∪ RxLy') → x * y = SNoCut (LxLy' ∪ RxRy') (LxRy' ∪ RxLy') → q) → q
In Proofgold the corresponding term root is 0c7af3... and proposition id is 32f888...
Proof:
The rest of this subproof is missing.
∎
Theorem. (mul_SNo_SNoCut_SNoL_interpolate)
∀Lx Rx Ly Ry, SNoCutP Lx Rx → SNoCutP Ly Ry → ∀x y, x = SNoCut Lx Rx → y = SNoCut Ly Ry → ∀u ∈ SNoL (x * y), (∃v ∈ Lx, ∃w ∈ Ly, u + v * w ≤ v * y + x * w) ∨ (∃v ∈ Rx, ∃w ∈ Ry, u + v * w ≤ v * y + x * w)
In Proofgold the corresponding term root is a1d92b... and proposition id is cc79b7...
Proof:
The rest of this subproof is missing.
∎
Theorem. (mul_SNo_SNoCut_SNoL_interpolate_impred)
∀Lx Rx Ly Ry, SNoCutP Lx Rx → SNoCutP Ly Ry → ∀x y, x = SNoCut Lx Rx → y = SNoCut Ly Ry → ∀u ∈ SNoL (x * y), ∀p : prop, (∀v ∈ Lx, ∀w ∈ Ly, u + v * w ≤ v * y + x * w → p) → (∀v ∈ Rx, ∀w ∈ Ry, u + v * w ≤ v * y + x * w → p) → p
In Proofgold the corresponding term root is 600e8b... and proposition id is f55f47...
Proof:
The rest of this subproof is missing.
∎
Theorem. (mul_SNo_SNoCut_SNoR_interpolate)
∀Lx Rx Ly Ry, SNoCutP Lx Rx → SNoCutP Ly Ry → ∀x y, x = SNoCut Lx Rx → y = SNoCut Ly Ry → ∀u ∈ SNoR (x * y), (∃v ∈ Lx, ∃w ∈ Ry, v * y + x * w ≤ u + v * w) ∨ (∃v ∈ Rx, ∃w ∈ Ly, v * y + x * w ≤ u + v * w)
In Proofgold the corresponding term root is 1f215f... and proposition id is a2fa89...
Proof:
The rest of this subproof is missing.
∎
Theorem. (mul_SNo_SNoCut_SNoR_interpolate_impred)
∀Lx Rx Ly Ry, SNoCutP Lx Rx → SNoCutP Ly Ry → ∀x y, x = SNoCut Lx Rx → y = SNoCut Ly Ry → ∀u ∈ SNoR (x * y), ∀p : prop, (∀v ∈ Lx, ∀w ∈ Ry, v * y + x * w ≤ u + v * w → p) → (∀v ∈ Rx, ∀w ∈ Ly, v * y + x * w ≤ u + v * w → p) → p
In Proofgold the corresponding term root is 544226... and proposition id is a2323a...
Proof:
The rest of this subproof is missing.
∎
Theorem. (nonpos_nonneg_0)
∀m n ∈ ω, m = - n → m = 0 ∧ n = 0
In Proofgold the corresponding term root is 94b9ef... and proposition id is bac05d...
Proof:
The rest of this subproof is missing.
∎
Theorem. (mul_minus_SNo_distrR)
∀x y, SNo x → SNo y → x * (- y) = - (x * y)
In Proofgold the corresponding term root is ab775a... and proposition id is 2e7475...
Proof:
The rest of this subproof is missing.
∎
End of Section SurrealMul
Beginning of Section Int
Notation. We use - as a prefix operator with priority 358 corresponding to applying term minus_SNo.
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_SNo.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_SNo.
Definition. We define int to be ω ∪ {- n|n ∈ ω} of type set.
In Proofgold the corresponding term root is 90ee85... and object id is 331669...
Theorem. (int_SNo_cases)
∀p : set → prop, (∀n ∈ ω, p n) → (∀n ∈ ω, p (- n)) → ∀x ∈ int, p x
In Proofgold the corresponding term root is 60107a... and proposition id is 14278d...
Proof:
The rest of this subproof is missing.
∎
Theorem. (int_3_cases)
∀n ∈ int, ∀p : prop, (∀m ∈ ω, n = - ordsucc m → p) → (n = 0 → p) → (∀m ∈ ω, n = ordsucc m → p) → p
In Proofgold the corresponding term root is 67b16d... and proposition id is 5c24db...
Proof:
The rest of this subproof is missing.
∎
Theorem. (int_SNo)
∀x ∈ int, SNo x
In Proofgold the corresponding term root is 19fdd2... and proposition id is 6089ec...
Proof:
The rest of this subproof is missing.
∎
Theorem. (Subq_omega_int)
ω ⊆ int
In Proofgold the corresponding term root is 405dda... and proposition id is b1bb2b...
Proof:
The rest of this subproof is missing.
∎
Theorem. (int_minus_SNo_omega)
∀n ∈ ω, - n ∈ int
In Proofgold the corresponding term root is 869503... and proposition id is 3a2593...
Proof:
The rest of this subproof is missing.
∎
Theorem. (int_add_SNo_lem)
∀n ∈ ω, ∀m, nat_p m → - n + m ∈ int
In Proofgold the corresponding term root is 8f0047... and proposition id is aab9d5...
Proof:
The rest of this subproof is missing.
∎
Theorem. (int_add_SNo)
∀x y ∈ int, x + y ∈ int
In Proofgold the corresponding term root is d66975... and proposition id is 1036d9...
Proof:
The rest of this subproof is missing.
∎
Theorem. (int_minus_SNo)
∀x ∈ int, - x ∈ int
In Proofgold the corresponding term root is 00e714... and proposition id is b2c591...
Proof:
The rest of this subproof is missing.
∎
Theorem. (int_mul_SNo)
∀x y ∈ int, x * y ∈ int
In Proofgold the corresponding term root is a7bc92... and proposition id is f40564...
Proof:
The rest of this subproof is missing.
∎
Theorem. (nonneg_int_nat_p)
∀n ∈ int, 0 ≤ n → nat_p n
In Proofgold the corresponding term root is 17e526... and proposition id is aa7e89...
Proof:
The rest of this subproof is missing.
∎
End of Section Int
Beginning of Section BezoutAndGCD
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_nat.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_nat.
Theorem. (quotient_remainder_nat)
∀n ∈ ω ∖ {0}, ∀m, nat_p m → ∃q ∈ ω, ∃r ∈ n, m = q * n + r
In Proofgold the corresponding term root is 9215c7... and proposition id is b1d852...
Proof:
The rest of this subproof is missing.
∎
Notation. We use - as a prefix operator with priority 358 corresponding to applying term minus_SNo.
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_SNo.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_SNo.
Notation. We use ^ as an infix operator with priority 342 and which associates to the right corresponding to applying term exp_SNo_nat.
Notation. We use < as an infix operator with priority 490 and no associativity corresponding to applying term SNoLt.
Notation. We use ≤ as an infix operator with priority 490 and no associativity corresponding to applying term SNoLe.
Theorem. (mul_SNo_nonpos_nonneg)
∀x y, SNo x → SNo y → x ≤ 0 → 0 ≤ y → x * y ≤ 0
In Proofgold the corresponding term root is be40cc... and proposition id is 07ca23...
Proof:
The rest of this subproof is missing.
∎
Theorem. (ordinal_0_In_ordsucc)
∀alpha, ordinal alpha → 0 ∈ ordsucc alpha
In Proofgold the corresponding term root is 62fdc5... and proposition id is a0377c...
Proof:
The rest of this subproof is missing.
∎
Theorem. (ordinal_ordsucc_pos)
∀alpha, ordinal alpha → 0 < ordsucc alpha
In Proofgold the corresponding term root is 9af5d3... and proposition id is 7e3b01...
Proof:
The rest of this subproof is missing.
∎
Theorem. (quotient_remainder_int)
∀n ∈ ω ∖ {0}, ∀m ∈ int, ∃q ∈ int, ∃r ∈ n, m = q * n + r
In Proofgold the corresponding term root is a18a83... and proposition id is 60c3e2...
Proof:
The rest of this subproof is missing.
∎
Definition. We define divides_int to be λm n ⇒ m ∈ int ∧ n ∈ int ∧ ∃k ∈ int, m * k = n of type set → set → prop.
In Proofgold the corresponding term root is c1f22f... and object id is ed2a3f...
Theorem. (divides_int_ref)
∀n ∈ int, divides_int n n
In Proofgold the corresponding term root is 6f5bb9... and proposition id is 5a3a8d...
Proof:
The rest of this subproof is missing.
∎
Theorem. (divides_int_0)
∀n ∈ int, divides_int n 0
In Proofgold the corresponding term root is f1256b... and proposition id is 0ec1b5...
Proof:
The rest of this subproof is missing.
∎
Theorem. (divides_int_add_SNo)
∀m n k, divides_int m n → divides_int m k → divides_int m (n + k)
In Proofgold the corresponding term root is 8466f3... and proposition id is 8cb5db...
Proof:
The rest of this subproof is missing.
∎
Theorem. (divides_int_mul_SNo)
∀m n m' n', divides_int m m' → divides_int n n' → divides_int (m * n) (m' * n')
In Proofgold the corresponding term root is 6a57ce... and proposition id is b51f19...
Proof:
The rest of this subproof is missing.
∎
Theorem. (divides_nat_divides_int)
∀m n, divides_nat m n → divides_int m n
In Proofgold the corresponding term root is c2b8d5... and proposition id is 5cfa58...
Proof:
The rest of this subproof is missing.
∎
Theorem. (divides_int_divides_nat)
∀m n ∈ ω, divides_int m n → divides_nat m n
In Proofgold the corresponding term root is 57df20... and proposition id is 6a6a50...
Proof:
The rest of this subproof is missing.
∎
Theorem. (divides_int_minus_SNo)
∀m n, divides_int m n → divides_int m (- n)
In Proofgold the corresponding term root is 28e67d... and proposition id is dcc59b...
Proof:
The rest of this subproof is missing.
∎
Theorem. (divides_int_mul_SNo_L)
∀m n, ∀k ∈ int, divides_int m n → divides_int m (n * k)
In Proofgold the corresponding term root is 202890... and proposition id is 1af430...
Proof:
The rest of this subproof is missing.
∎
Theorem. (divides_int_mul_SNo_R)
∀m n, ∀k ∈ int, divides_int m n → divides_int m (k * n)
In Proofgold the corresponding term root is 08581b... and proposition id is 573697...
Proof:
The rest of this subproof is missing.
∎
Theorem. (divides_int_1)
∀n ∈ int, divides_int 1 n
In Proofgold the corresponding term root is c86e00... and proposition id is cb6ed6...
Proof:
The rest of this subproof is missing.
∎
Theorem. (divides_int_pos_Le)
∀m n, divides_int m n → 0 < n → m ≤ n
In Proofgold the corresponding term root is 310556... and proposition id is 586793...
Proof:
The rest of this subproof is missing.
∎
Definition. We define gcd_reln to be λm n d ⇒ divides_int d m ∧ divides_int d n ∧ ∀d', divides_int d' m → divides_int d' n → d' ≤ d of type set → set → set → prop.
In Proofgold the corresponding term root is 9ed1e3... and object id is 5f6921...
Theorem. (gcd_reln_uniq)
∀a b c d, gcd_reln a b c → gcd_reln a b d → c = d
In Proofgold the corresponding term root is 77eeb9... and proposition id is 2e39ac...
Proof:
The rest of this subproof is missing.
∎
Definition. We define int_lin_comb to be λa b c ⇒ a ∈ int ∧ b ∈ int ∧ c ∈ int ∧ ∃m n ∈ int, m * a + n * b = c of type set → set → set → prop.
In Proofgold the corresponding term root is 5ca407... and object id is 46c13c...
Theorem. (int_lin_comb_I)
∀a b c ∈ int, (∃m n ∈ int, m * a + n * b = c) → int_lin_comb a b c
In Proofgold the corresponding term root is ee0af1... and proposition id is c74b8d...
Proof:
The rest of this subproof is missing.
∎
Theorem. (int_lin_comb_E)
∀a b c, int_lin_comb a b c → ∀p : prop, (a ∈ int → b ∈ int → c ∈ int → ∀m n ∈ int, m * a + n * b = c → p) → p
In Proofgold the corresponding term root is c38ec0... and proposition id is f9a364...
Proof:
The rest of this subproof is missing.
∎
Theorem. (int_lin_comb_E1)
∀a b c, int_lin_comb a b c → a ∈ int
In Proofgold the corresponding term root is 8760f2... and proposition id is 607ef2...
Proof:
The rest of this subproof is missing.
∎
Theorem. (int_lin_comb_E2)
∀a b c, int_lin_comb a b c → b ∈ int
In Proofgold the corresponding term root is d61e74... and proposition id is 239909...
Proof:
The rest of this subproof is missing.
∎
Theorem. (int_lin_comb_E3)
∀a b c, int_lin_comb a b c → c ∈ int
In Proofgold the corresponding term root is 2d6f43... and proposition id is 4114d5...
Proof:
The rest of this subproof is missing.
∎
Theorem. (int_lin_comb_E4)
∀a b c, int_lin_comb a b c → ∀p : prop, (∀m n ∈ int, m * a + n * b = c → p) → p
In Proofgold the corresponding term root is a419d4... and proposition id is 7b482f...
Proof:
The rest of this subproof is missing.
∎
Theorem. (least_pos_int_lin_comb_ex)
∀a b ∈ int, ¬ (a = 0 ∧ b = 0) → ∃c, int_lin_comb a b c ∧ 0 < c ∧ ∀c', int_lin_comb a b c' → 0 < c' → c ≤ c'
In Proofgold the corresponding term root is adb37d... and proposition id is 3821bd...
Proof:
The rest of this subproof is missing.
∎
Theorem. (int_lin_comb_sym)
∀a b d, int_lin_comb a b d → int_lin_comb b a d
In Proofgold the corresponding term root is 78fc6f... and proposition id is 75991b...
Proof:
The rest of this subproof is missing.
∎
Theorem. (least_pos_int_lin_comb_divides_int)
∀a b d, int_lin_comb a b d → 0 < d → (∀c, int_lin_comb a b c → 0 < c → d ≤ c) → divides_int d a
In Proofgold the corresponding term root is f354de... and proposition id is 68c498...
Proof:
The rest of this subproof is missing.
∎
Theorem. (least_pos_int_lin_comb_gcd)
∀a b d, int_lin_comb a b d → 0 < d → (∀c, int_lin_comb a b c → 0 < c → d ≤ c) → gcd_reln a b d
In Proofgold the corresponding term root is 9c320d... and proposition id is f5a534...
Proof:
The rest of this subproof is missing.
∎
Theorem. (BezoutThm)
∀a b ∈ int, ¬ (a = 0 ∧ b = 0) → ∀d, gcd_reln a b d ↔ int_lin_comb a b d ∧ 0 < d ∧ ∀d', int_lin_comb a b d' → 0 < d' → d ≤ d'
In Proofgold the corresponding term root is 01513f... and proposition id is ed22be...
Proof:
The rest of this subproof is missing.
∎
Theorem. (gcd_id)
∀m ∈ ω ∖ {0}, gcd_reln m m m
In Proofgold the corresponding term root is 97c37f... and proposition id is 4d0b43...
Proof:
The rest of this subproof is missing.
∎
Theorem. (gcd_0)
∀m ∈ ω ∖ {0}, gcd_reln 0 m m
In Proofgold the corresponding term root is a22353... and proposition id is f93fc3...
Proof:
The rest of this subproof is missing.
∎
Theorem. (gcd_sym)
∀m n d, gcd_reln m n d → gcd_reln n m d
In Proofgold the corresponding term root is 0a399e... and proposition id is ac8b69...
Proof:
The rest of this subproof is missing.
∎
Theorem. (gcd_minus)
∀m n d, gcd_reln m n d → gcd_reln m (- n) d
In Proofgold the corresponding term root is 261ac3... and proposition id is d054b8...
Proof:
The rest of this subproof is missing.
∎
Theorem. (euclidean_algorithm_prop_1)
∀m n d, n ∈ int → gcd_reln m (n + - m) d → gcd_reln m n d
In Proofgold the corresponding term root is a6b8d4... and proposition id is 29c3b8...
Proof:
The rest of this subproof is missing.
∎
Theorem. (euclidean_algorithm)
(∀m ∈ ω ∖ {0}, gcd_reln m m m) ∧ (∀m ∈ ω ∖ {0}, gcd_reln 0 m m) ∧ (∀m ∈ ω ∖ {0}, gcd_reln m 0 m) ∧ (∀m n ∈ ω, m < n → ∀d, gcd_reln m (n + - m) d → gcd_reln m n d) ∧ (∀m n ∈ ω, n < m → ∀d, gcd_reln n m d → gcd_reln m n d) ∧ (∀m ∈ ω, ∀n ∈ int, n < 0 → ∀d, gcd_reln m (- n) d → gcd_reln m n d) ∧ (∀m n ∈ int, m < 0 → ∀d, gcd_reln (- m) n d → gcd_reln m n d)
In Proofgold the corresponding term root is d87f07... and proposition id is 24974e...
Proof:
The rest of this subproof is missing.
∎
Theorem. (Euclid_lemma)
∀p, prime_nat p → ∀a b ∈ int, divides_int p (a * b) → divides_int p a ∨ divides_int p b
In Proofgold the corresponding term root is 786cc5... and proposition id is 2d187a...
Proof:
The rest of this subproof is missing.
∎
End of Section BezoutAndGCD
Beginning of Section PrimeFactorization
Notation. We use - as a prefix operator with priority 358 corresponding to applying term minus_SNo.
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_SNo.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_SNo.
Notation. We use ^ as an infix operator with priority 342 and which associates to the right corresponding to applying term exp_SNo_nat.
Notation. We use < as an infix operator with priority 490 and no associativity corresponding to applying term SNoLt.
Notation. We use ≤ as an infix operator with priority 490 and no associativity corresponding to applying term SNoLe.
Theorem. (prime_not_divides_int_1)
∀p, prime_nat p → ¬ divides_int p 1
In Proofgold the corresponding term root is b9f761... and proposition id is 10be61...
Proof:
The rest of this subproof is missing.
∎
Definition. We define Pi_SNo to be λf n ⇒ nat_primrec 1 (λi r ⇒ r * f i) n of type (set → set) → set → set.
In Proofgold the corresponding term root is 3e6f6f... and object id is 05ecb8...
Theorem. (Pi_SNo_0)
∀f : set → set, Pi_SNo f 0 = 1
In Proofgold the corresponding term root is 89e731... and proposition id is b6e187...
Proof:
The rest of this subproof is missing.
∎
Theorem. (Pi_SNo_S)
∀f : set → set, ∀n, nat_p n → Pi_SNo f (ordsucc n) = Pi_SNo f n * f n
In Proofgold the corresponding term root is bfe386... and proposition id is 33475e...
Proof:
The rest of this subproof is missing.
∎
Theorem. (Pi_SNo_In_omega)
∀f : set → set, ∀n, nat_p n → (∀i ∈ n, f i ∈ ω) → Pi_SNo f n ∈ ω
In Proofgold the corresponding term root is b7aa6a... and proposition id is 63bfef...
Proof:
The rest of this subproof is missing.
∎
Theorem. (Pi_SNo_In_int)
∀f : set → set, ∀n, nat_p n → (∀i ∈ n, f i ∈ int) → Pi_SNo f n ∈ int
In Proofgold the corresponding term root is 011f99... and proposition id is 22c84f...
Proof:
The rest of this subproof is missing.
∎
Theorem. (divides_int_prime_nat_eq)
∀p q, prime_nat p → prime_nat q → divides_int p q → p = q
In Proofgold the corresponding term root is 5e30a7... and proposition id is d07df4...
Proof:
The rest of this subproof is missing.
∎
Theorem. (Euclid_lemma_Pi_SNo)
∀f : set → set, ∀p, prime_nat p → ∀n, nat_p n → (∀i ∈ n, f i ∈ int) → divides_int p (Pi_SNo f n) → ∃i ∈ n, divides_int p (f i)
In Proofgold the corresponding term root is 50a15b... and proposition id is 2326d1...
Proof:
The rest of this subproof is missing.
∎
Theorem. (divides_nat_mul_SNo_R)
∀m n ∈ ω, divides_nat m (m * n)
In Proofgold the corresponding term root is 68b6fa... and proposition id is a297e7...
Proof:
The rest of this subproof is missing.
∎
Theorem. (divides_nat_mul_SNo_L)
∀m n ∈ ω, divides_nat n (m * n)
In Proofgold the corresponding term root is 1f5742... and proposition id is a0cc9d...
Proof:
The rest of this subproof is missing.
∎
Theorem. (Pi_SNo_divides)
∀f : set → set, ∀n, nat_p n → (∀i ∈ n, f i ∈ ω) → (∀i ∈ n, divides_nat (f i) (Pi_SNo f n))
In Proofgold the corresponding term root is dd587f... and proposition id is 3899cc...
Proof:
The rest of this subproof is missing.
∎
Definition. We define nonincrfinseq to be λA n f ⇒ ∀i ∈ n, A (f i) ∧ ∀j ∈ i, f i ≤ f j of type (set → prop) → set → (set → set) → prop.
In Proofgold the corresponding term root is f77d44... and object id is 168aa7...
Theorem. (Pi_SNo_eq)
∀f g : set → set, ∀m, nat_p m → (∀i ∈ m, f i = g i) → Pi_SNo f m = Pi_SNo g m
In Proofgold the corresponding term root is c1ba17... and proposition id is 9a5d69...
Proof:
The rest of this subproof is missing.
∎
Theorem. (prime_factorization_ex_uniq)
∀n, nat_p n → 0 ∈ n → ∃k ∈ ω, ∃f : set → set, nonincrfinseq prime_nat k f ∧ Pi_SNo f k = n ∧ ∀k' ∈ ω, ∀f' : set → set, nonincrfinseq prime_nat k' f' → Pi_SNo f' k' = n → k' = k ∧ ∀i ∈ k, f' i = f i
In Proofgold the corresponding term root is ae2a40... and proposition id is a96cd8...
Proof:
The rest of this subproof is missing.
∎
End of Section PrimeFactorization
Beginning of Section SurrealExp
Notation. We use - as a prefix operator with priority 358 corresponding to applying term minus_SNo.
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_SNo.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_SNo.
Definition. We define exp_SNo_nat to be λn m : set ⇒ nat_primrec 1 (λ_ r ⇒ n * r) m of type set → set → set.
In Proofgold the corresponding term root is cc5143... and object id is ea8104...
Notation. We use ^ as an infix operator with priority 342 and which associates to the right corresponding to applying term exp_SNo_nat.
Theorem. (exp_SNo_nat_0)
∀x, SNo x → x ^ 0 = 1
In Proofgold the corresponding term root is c266f2... and proposition id is fe30fb...
Proof:
The rest of this subproof is missing.
∎
Theorem. (exp_SNo_nat_S)
∀x, SNo x → ∀n, nat_p n → x ^ (ordsucc n) = x * x ^ n
In Proofgold the corresponding term root is 6b5ef9... and proposition id is ae55b7...
Proof:
The rest of this subproof is missing.
∎
Theorem. (exp_SNo_nat_1)
∀x, SNo x → x ^ 1 = x
In Proofgold the corresponding term root is 9a6103... and proposition id is d52b56...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNo_exp_SNo_nat)
∀x, SNo x → ∀n, nat_p n → SNo (x ^ n)
In Proofgold the corresponding term root is 7a77b4... and proposition id is f317bf...
Proof:
The rest of this subproof is missing.
∎
Theorem. (nat_exp_SNo_nat)
∀x, nat_p x → ∀n, nat_p n → nat_p (x ^ n)
In Proofgold the corresponding term root is 14de73... and proposition id is 39327e...
Proof:
The rest of this subproof is missing.
∎
Theorem. (eps_ordsucc_half_add)
∀n, nat_p n → eps_ (ordsucc n) + eps_ (ordsucc n) = eps_ n
In Proofgold the corresponding term root is 6f06cd... and proposition id is 1a029d...
Proof:
The rest of this subproof is missing.
∎
Theorem. (eps_1_half_eq1)
eps_ 1 + eps_ 1 = 1
In Proofgold the corresponding term root is 356e75... and proposition id is 2163a7...
Proof:
The rest of this subproof is missing.
∎
Theorem. (eps_1_half_eq2)
2 * eps_ 1 = 1
In Proofgold the corresponding term root is 69f0c6... and proposition id is 95d3cb...
Proof:
The rest of this subproof is missing.
∎
Theorem. (double_eps_1)
∀x y z, SNo x → SNo y → SNo z → x + x = y + z → x = eps_ 1 * (y + z)
In Proofgold the corresponding term root is 42def0... and proposition id is 896286...
Proof:
The rest of this subproof is missing.
∎
Theorem. (exp_SNo_1_bd)
∀x, SNo x → 1 ≤ x → ∀n, nat_p n → 1 ≤ x ^ n
In Proofgold the corresponding term root is a5c549... and proposition id is 002e5b...
Proof:
The rest of this subproof is missing.
∎
Theorem. (exp_SNo_2_bd)
∀n, nat_p n → n < 2 ^ n
In Proofgold the corresponding term root is 1297d1... and proposition id is 88dd94...
Proof:
The rest of this subproof is missing.
∎
Theorem. (mul_SNo_eps_power_2)
∀n, nat_p n → eps_ n * 2 ^ n = 1
In Proofgold the corresponding term root is 6921ff... and proposition id is e69e44...
Proof:
The rest of this subproof is missing.
∎
Theorem. (eps_bd_1)
∀n ∈ ω, eps_ n ≤ 1
In Proofgold the corresponding term root is b781ab... and proposition id is 1165c4...
Proof:
The rest of this subproof is missing.
∎
Theorem. (mul_SNo_eps_power_2')
∀n, nat_p n → 2 ^ n * eps_ n = 1
In Proofgold the corresponding term root is 721eb5... and proposition id is ce9559...
Proof:
The rest of this subproof is missing.
∎
Theorem. (exp_SNo_nat_mul_add)
∀x, SNo x → ∀m, nat_p m → ∀n, nat_p n → x ^ m * x ^ n = x ^ (m + n)
In Proofgold the corresponding term root is 3333a9... and proposition id is f04547...
Proof:
The rest of this subproof is missing.
∎
Theorem. (exp_SNo_nat_mul_add')
∀x, SNo x → ∀m n ∈ ω, x ^ m * x ^ n = x ^ (m + n)
In Proofgold the corresponding term root is 364123... and proposition id is bde099...
Proof:
The rest of this subproof is missing.
∎
Theorem. (exp_SNo_nat_pos)
∀x, SNo x → 0 < x → ∀n, nat_p n → 0 < x ^ n
In Proofgold the corresponding term root is 795806... and proposition id is 6538ad...
Proof:
The rest of this subproof is missing.
∎
Theorem. (mul_SNo_eps_eps_add_SNo)
∀m n ∈ ω, eps_ m * eps_ n = eps_ (m + n)
In Proofgold the corresponding term root is 857b85... and proposition id is ea6877...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNoS_omega_Lev_equip)
∀n, nat_p n → equip {x ∈ SNoS_ ω|SNoLev x = n} (2 ^ n)
In Proofgold the corresponding term root is 2712ea... and proposition id is 81ef79...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNoS_finite)
∀n ∈ ω, finite (SNoS_ n)
In Proofgold the corresponding term root is b18610... and proposition id is c5538e...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNoS_omega_SNoL_finite)
∀x ∈ SNoS_ ω, finite (SNoL x)
In Proofgold the corresponding term root is bf4159... and proposition id is 473468...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNoS_omega_SNoR_finite)
∀x ∈ SNoS_ ω, finite (SNoR x)
In Proofgold the corresponding term root is c352d0... and proposition id is eafcc9...
Proof:
The rest of this subproof is missing.
∎
End of Section SurrealExp
Beginning of Section SNoMaxMin
Notation. We use - as a prefix operator with priority 358 corresponding to applying term minus_SNo.
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_SNo.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_SNo.
Notation. We use ^ as an infix operator with priority 342 and which associates to the right corresponding to applying term exp_SNo_nat.
Notation. We use < as an infix operator with priority 490 and no associativity corresponding to applying term SNoLt.
Notation. We use ≤ as an infix operator with priority 490 and no associativity corresponding to applying term SNoLe.
Definition. We define SNo_max_of to be λX x ⇒ x ∈ X ∧ SNo x ∧ ∀y ∈ X, SNo y → y ≤ x of type set → set → prop.
In Proofgold the corresponding term root is 9570d4... and object id is 0e9f0e...
Definition. We define SNo_min_of to be λX x ⇒ x ∈ X ∧ SNo x ∧ ∀y ∈ X, SNo y → x ≤ y of type set → set → prop.
In Proofgold the corresponding term root is 5eb8d7... and object id is 144946...
Theorem. (minus_SNo_max_min)
∀X y, (∀x ∈ X, SNo x) → SNo_max_of X y → SNo_min_of {- x|x ∈ X} (- y)
In Proofgold the corresponding term root is 2403cf... and proposition id is f6c519...
Proof:
The rest of this subproof is missing.
∎
Theorem. (minus_SNo_max_min')
∀X y, (∀x ∈ X, SNo x) → SNo_max_of {- x|x ∈ X} y → SNo_min_of X (- y)
In Proofgold the corresponding term root is 544581... and proposition id is b07737...
Proof:
The rest of this subproof is missing.
∎
Theorem. (minus_SNo_min_max)
∀X y, (∀x ∈ X, SNo x) → SNo_min_of X y → SNo_max_of {- x|x ∈ X} (- y)
In Proofgold the corresponding term root is d435c2... and proposition id is fbf599...
Proof:
The rest of this subproof is missing.
∎
Theorem. (double_SNo_max_1)
∀x y, SNo x → SNo_max_of (SNoL x) y → ∀z, SNo z → x < z → y + z < x + x → ∃w ∈ SNoR z, y + w = x + x
In Proofgold the corresponding term root is 3155e8... and proposition id is d66034...
Proof:
The rest of this subproof is missing.
∎
Theorem. (double_SNo_min_1)
∀x y, SNo x → SNo_min_of (SNoR x) y → ∀z, SNo z → z < x → x + x < y + z → ∃w ∈ SNoL z, y + w = x + x
In Proofgold the corresponding term root is fa7d8b... and proposition id is 084f04...
Proof:
The rest of this subproof is missing.
∎
Theorem. (finite_max_exists)
∀X, (∀x ∈ X, SNo x) → finite X → X ≠ 0 → ∃x, SNo_max_of X x
In Proofgold the corresponding term root is 182021... and proposition id is 9baa0b...
Proof:
The rest of this subproof is missing.
∎
Theorem. (finite_min_exists)
∀X, (∀x ∈ X, SNo x) → finite X → X ≠ 0 → ∃x, SNo_min_of X x
In Proofgold the corresponding term root is efc9f1... and proposition id is 25808a...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNoS_omega_SNoL_max_exists)
∀x ∈ SNoS_ ω, SNoL x = 0 ∨ ∃y, SNo_max_of (SNoL x) y
In Proofgold the corresponding term root is 4539ed... and proposition id is faaae1...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNoS_omega_SNoR_min_exists)
∀x ∈ SNoS_ ω, SNoR x = 0 ∨ ∃y, SNo_min_of (SNoR x) y
In Proofgold the corresponding term root is b0aa0c... and proposition id is baaae8...
Proof:
The rest of this subproof is missing.
∎
End of Section SNoMaxMin
Beginning of Section DiadicRationals
Notation. We use - as a prefix operator with priority 358 corresponding to applying term minus_SNo.
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_SNo.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_SNo.
Notation. We use < as an infix operator with priority 490 and no associativity corresponding to applying term SNoLt.
Notation. We use ≤ as an infix operator with priority 490 and no associativity corresponding to applying term SNoLe.
Notation. We use ^ as an infix operator with priority 342 and which associates to the right corresponding to applying term exp_SNo_nat.
Theorem. (nonneg_diadic_rational_p_SNoS_omega)
∀k ∈ ω, ∀n, nat_p n → eps_ k * n ∈ SNoS_ ω
In Proofgold the corresponding term root is 9323b5... and proposition id is d9b318...
Proof:
The rest of this subproof is missing.
∎
Definition. We define diadic_rational_p to be λx ⇒ ∃k ∈ ω, ∃m ∈ int, x = eps_ k * m of type set → prop.
In Proofgold the corresponding term root is a02ca9... and object id is 25bca1...
Theorem. (diadic_rational_p_SNoS_omega)
∀x, diadic_rational_p x → x ∈ SNoS_ ω
In Proofgold the corresponding term root is b7e7af... and proposition id is 3dea67...
Proof:
The rest of this subproof is missing.
∎
Theorem. (int_diadic_rational_p)
∀m ∈ int, diadic_rational_p m
In Proofgold the corresponding term root is 88ce9c... and proposition id is 8a4cf7...
Proof:
The rest of this subproof is missing.
∎
Theorem. (omega_diadic_rational_p)
∀m ∈ ω, diadic_rational_p m
In Proofgold the corresponding term root is 40b048... and proposition id is c250b0...
Proof:
The rest of this subproof is missing.
∎
Theorem. (eps_diadic_rational_p)
∀k ∈ ω, diadic_rational_p (eps_ k)
In Proofgold the corresponding term root is d306dc... and proposition id is cc1098...
Proof:
The rest of this subproof is missing.
∎
Theorem. (minus_SNo_diadic_rational_p)
∀x, diadic_rational_p x → diadic_rational_p (- x)
In Proofgold the corresponding term root is bb6f7d... and proposition id is bc4825...
Proof:
The rest of this subproof is missing.
∎
Theorem. (mul_SNo_diadic_rational_p)
∀x y, diadic_rational_p x → diadic_rational_p y → diadic_rational_p (x * y)
In Proofgold the corresponding term root is ed6a37... and proposition id is e4ff6e...
Proof:
The rest of this subproof is missing.
∎
Theorem. (add_SNo_diadic_rational_p)
∀x y, diadic_rational_p x → diadic_rational_p y → diadic_rational_p (x + y)
In Proofgold the corresponding term root is bd7ab2... and proposition id is 4e6254...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNoS_omega_diadic_rational_p_lem)
∀n, nat_p n → ∀x, SNo x → SNoLev x = n → diadic_rational_p x
In Proofgold the corresponding term root is 85a5ea... and proposition id is bf97d9...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNoS_omega_diadic_rational_p)
∀x ∈ SNoS_ ω, diadic_rational_p x
In Proofgold the corresponding term root is a9c764... and proposition id is 963bd5...
Proof:
The rest of this subproof is missing.
∎
Theorem. (mul_SNo_SNoS_omega)
∀x y ∈ SNoS_ ω, x * y ∈ SNoS_ ω
In Proofgold the corresponding term root is d17aa1... and proposition id is 768371...
Proof:
The rest of this subproof is missing.
∎
End of Section DiadicRationals
Beginning of Section SurrealDiv
Notation. We use - as a prefix operator with priority 358 corresponding to applying term minus_SNo.
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_SNo.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_SNo.
Notation. We use < as an infix operator with priority 490 and no associativity corresponding to applying term SNoLt.
Notation. We use ≤ as an infix operator with priority 490 and no associativity corresponding to applying term SNoLe.
Notation. We use ^ as an infix operator with priority 342 and which associates to the right corresponding to applying term exp_SNo_nat.
Definition. We define SNoL_pos to be λx ⇒ {w ∈ SNoL x|0 < w} of type set → set.
In Proofgold the corresponding term root is 399f06... and object id is 65d5ad...
Theorem. (SNo_recip_pos_pos)
∀x xi, SNo x → SNo xi → 0 < x → x * xi = 1 → 0 < xi
In Proofgold the corresponding term root is 8f0063... and proposition id is 5b48db...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNo_recip_lem1)
∀x x' x'i y y', SNo x → 0 < x → x' ∈ SNoL_pos x → SNo x'i → x' * x'i = 1 → SNo y → x * y < 1 → SNo y' → 1 + - x * y' = (1 + - x * y) * (x' + - x) * x'i → 1 < x * y'
In Proofgold the corresponding term root is 761151... and proposition id is b197c8...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNo_recip_lem2)
∀x x' x'i y y', SNo x → 0 < x → x' ∈ SNoL_pos x → SNo x'i → x' * x'i = 1 → SNo y → 1 < x * y → SNo y' → 1 + - x * y' = (1 + - x * y) * (x' + - x) * x'i → x * y' < 1
In Proofgold the corresponding term root is 423150... and proposition id is 1a06f3...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNo_recip_lem3)
∀x x' x'i y y', SNo x → 0 < x → x' ∈ SNoR x → SNo x'i → x' * x'i = 1 → SNo y → x * y < 1 → SNo y' → 1 + - x * y' = (1 + - x * y) * (x' + - x) * x'i → x * y' < 1
In Proofgold the corresponding term root is b6e635... and proposition id is 22668c...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNo_recip_lem4)
∀x x' x'i y y', SNo x → 0 < x → x' ∈ SNoR x → SNo x'i → x' * x'i = 1 → SNo y → 1 < x * y → SNo y' → 1 + - x * y' = (1 + - x * y) * (x' + - x) * x'i → 1 < x * y'
In Proofgold the corresponding term root is 417306... and proposition id is b572bc...
Proof:
The rest of this subproof is missing.
∎
Definition. We define SNo_recipauxset to be λY x X g ⇒ ⋃y ∈ Y{(1 + (x' + - x) * y) * g x'|x' ∈ X} of type set → set → set → (set → set) → set.
In Proofgold the corresponding term root is f6355f... and object id is d40943...
Theorem. (SNo_recipauxset_I)
∀Y x X, ∀g : set → set, ∀y ∈ Y, ∀x' ∈ X, (1 + (x' + - x) * y) * g x' ∈ SNo_recipauxset Y x X g
In Proofgold the corresponding term root is fb23ea... and proposition id is 3cd34a...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNo_recipauxset_E)
∀Y x X, ∀g : set → set, ∀z ∈ SNo_recipauxset Y x X g, ∀p : prop, (∀y ∈ Y, ∀x' ∈ X, z = (1 + (x' + - x) * y) * g x' → p) → p
In Proofgold the corresponding term root is ea0c30... and proposition id is 9c174a...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNo_recipauxset_ext)
∀Y x X, ∀g h : set → set, (∀x' ∈ X, g x' = h x') → SNo_recipauxset Y x X g = SNo_recipauxset Y x X h
In Proofgold the corresponding term root is 7eb20f... and proposition id is a033a6...
Proof:
The rest of this subproof is missing.
∎
Definition. We define SNo_recipaux to be λx g ⇒ nat_primrec ({0},0) (λk p ⇒ (p 0 ∪ SNo_recipauxset (p 0) x (SNoR x) g ∪ SNo_recipauxset (p 1) x (SNoL_pos x) g,p 1 ∪ SNo_recipauxset (p 0) x (SNoL_pos x) g ∪ SNo_recipauxset (p 1) x (SNoR x) g)) of type set → (set → set) → set → set.
In Proofgold the corresponding term root is 8cdbea... and object id is 6fc28a...
Theorem. (SNo_recipaux_0)
∀x, ∀g : set → set, SNo_recipaux x g 0 = ({0},0)
In Proofgold the corresponding term root is 1a1104... and proposition id is a9fc4d...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNo_recipaux_S)
∀x, ∀g : set → set, ∀n, nat_p n → SNo_recipaux x g (ordsucc n) = (SNo_recipaux x g n 0 ∪ SNo_recipauxset (SNo_recipaux x g n 0) x (SNoR x) g ∪ SNo_recipauxset (SNo_recipaux x g n 1) x (SNoL_pos x) g,SNo_recipaux x g n 1 ∪ SNo_recipauxset (SNo_recipaux x g n 0) x (SNoL_pos x) g ∪ SNo_recipauxset (SNo_recipaux x g n 1) x (SNoR x) g)
In Proofgold the corresponding term root is 35af7d... and proposition id is 55b106...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNo_recipaux_lem1)
∀x, SNo x → 0 < x → ∀g : set → set, (∀x' ∈ SNoS_ (SNoLev x), 0 < x' → SNo (g x') ∧ x' * g x' = 1) → ∀k, nat_p k → (∀y ∈ SNo_recipaux x g k 0, SNo y ∧ x * y < 1) ∧ (∀y ∈ SNo_recipaux x g k 1, SNo y ∧ 1 < x * y)
In Proofgold the corresponding term root is af615d... and proposition id is 7c3309...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNo_recipaux_lem2)
∀x, SNo x → 0 < x → ∀g : set → set, (∀x' ∈ SNoS_ (SNoLev x), 0 < x' → SNo (g x') ∧ x' * g x' = 1) → SNoCutP (⋃k ∈ ωSNo_recipaux x g k 0) (⋃k ∈ ωSNo_recipaux x g k 1)
In Proofgold the corresponding term root is 38996c... and proposition id is c6f6db...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNo_recipaux_ext)
∀x, SNo x → ∀g h : set → set, (∀x' ∈ SNoS_ (SNoLev x), g x' = h x') → ∀k, nat_p k → SNo_recipaux x g k = SNo_recipaux x h k
In Proofgold the corresponding term root is b2de41... and proposition id is e7b024...
Proof:
The rest of this subproof is missing.
∎
Beginning of Section recip_SNo_pos
Let G : set → (set → set) → set ≝ λx g ⇒ SNoCut (⋃k ∈ ωSNo_recipaux x g k 0) (⋃k ∈ ωSNo_recipaux x g k 1)
Definition. We define recip_SNo_pos to be SNo_rec_i G of type set → set.
In Proofgold the corresponding term root is 88641a... and object id is 2f4ee1...
Theorem. (recip_SNo_pos_eq)
∀x, SNo x → recip_SNo_pos x = G x recip_SNo_pos
In Proofgold the corresponding term root is 2e3e0e... and proposition id is ebab77...
Proof:
The rest of this subproof is missing.
∎
Theorem. (recip_SNo_pos_prop1)
∀x, SNo x → 0 < x → SNo (recip_SNo_pos x) ∧ x * recip_SNo_pos x = 1
In Proofgold the corresponding term root is 42138c... and proposition id is 785262...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNo_recip_SNo_pos)
∀x, SNo x → 0 < x → SNo (recip_SNo_pos x)
In Proofgold the corresponding term root is 90e6e6... and proposition id is 6b9294...
Proof:
The rest of this subproof is missing.
∎
Theorem. (recip_SNo_pos_invR)
∀x, SNo x → 0 < x → x * recip_SNo_pos x = 1
In Proofgold the corresponding term root is 12c412... and proposition id is a4f83c...
Proof:
The rest of this subproof is missing.
∎
Theorem. (recip_SNo_pos_is_pos)
∀x, SNo x → 0 < x → 0 < recip_SNo_pos x
In Proofgold the corresponding term root is c95744... and proposition id is 30d6a5...
Proof:
The rest of this subproof is missing.
∎
Theorem. (recip_SNo_pos_invol)
∀x, SNo x → 0 < x → recip_SNo_pos (recip_SNo_pos x) = x
In Proofgold the corresponding term root is 5e7409... and proposition id is a3c6c5...
Proof:
The rest of this subproof is missing.
∎
Theorem. (recip_SNo_pos_eps_)
∀n, nat_p n → recip_SNo_pos (eps_ n) = 2 ^ n
In Proofgold the corresponding term root is ce7e5d... and proposition id is 3c7219...
Proof:
The rest of this subproof is missing.
∎
Theorem. (recip_SNo_pos_pow_2)
∀n, nat_p n → recip_SNo_pos (2 ^ n) = eps_ n
In Proofgold the corresponding term root is e4f02a... and proposition id is 9ca0e9...
Proof:
The rest of this subproof is missing.
∎
Theorem. (recip_SNo_pos_2)
recip_SNo_pos 2 = eps_ 1
In Proofgold the corresponding term root is 1953c2... and proposition id is 4addd0...
Proof:
The rest of this subproof is missing.
∎
End of Section recip_SNo_pos
Definition. We define recip_SNo to be λx ⇒ if 0 < x then recip_SNo_pos x else if x < 0 then - recip_SNo_pos (- x) else 0 of type set → set.
In Proofgold the corresponding term root is aa96a2... and object id is 0d39b7...
Theorem. (recip_SNo_poscase)
∀x, 0 < x → recip_SNo x = recip_SNo_pos x
In Proofgold the corresponding term root is 152688... and proposition id is 2f43fd...
Proof:
The rest of this subproof is missing.
∎
Theorem. (recip_SNo_negcase)
∀x, SNo x → x < 0 → recip_SNo x = - recip_SNo_pos (- x)
In Proofgold the corresponding term root is c96182... and proposition id is 126cc9...
Proof:
The rest of this subproof is missing.
∎
Theorem. (recip_SNo_0)
recip_SNo 0 = 0
In Proofgold the corresponding term root is bda1b2... and proposition id is 466dab...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNo_recip_SNo)
∀x, SNo x → SNo (recip_SNo x)
In Proofgold the corresponding term root is 675572... and proposition id is 9f0369...
Proof:
The rest of this subproof is missing.
∎
Theorem. (recip_SNo_invR)
∀x, SNo x → x ≠ 0 → x * recip_SNo x = 1
In Proofgold the corresponding term root is 33c26e... and proposition id is 446f16...
Proof:
The rest of this subproof is missing.
∎
Theorem. (recip_SNo_invL)
∀x, SNo x → x ≠ 0 → recip_SNo x * x = 1
In Proofgold the corresponding term root is 85d435... and proposition id is e80fdb...
Proof:
The rest of this subproof is missing.
∎
Theorem. (mul_SNo_nonzero_cancel_L)
∀x y z, SNo x → x ≠ 0 → SNo y → SNo z → x * y = x * z → y = z
In Proofgold the corresponding term root is 67c26c... and proposition id is 035c5c...
Proof:
The rest of this subproof is missing.
∎
Theorem. (recip_SNo_pow_2)
∀n, nat_p n → recip_SNo (2 ^ n) = eps_ n
In Proofgold the corresponding term root is 761b63... and proposition id is 80f328...
Proof:
The rest of this subproof is missing.
∎
Theorem. (recip_SNo_of_pos_is_pos)
∀x, SNo x → 0 < x → 0 < recip_SNo x
In Proofgold the corresponding term root is 8f5692... and proposition id is 1015d4...
Proof:
The rest of this subproof is missing.
∎
Definition. We define div_SNo to be λx y ⇒ x * recip_SNo y of type set → set → set.
In Proofgold the corresponding term root is a77cc8... and object id is 8b12fd...
Notation. We use :/: as an infix operator with priority 353 and no associativity corresponding to applying term div_SNo.
Theorem. (SNo_div_SNo)
∀x y, SNo x → SNo y → SNo (x :/: y)
In Proofgold the corresponding term root is cebe75... and proposition id is c761b5...
Proof:
The rest of this subproof is missing.
∎
Theorem. (div_SNo_0_num)
∀x, SNo x → 0 :/: x = 0
In Proofgold the corresponding term root is 28c3dc... and proposition id is ec5f74...
Proof:
The rest of this subproof is missing.
∎
Theorem. (div_SNo_0_denum)
∀x, SNo x → x :/: 0 = 0
In Proofgold the corresponding term root is d6eb4e... and proposition id is 46b3d2...
Proof:
The rest of this subproof is missing.
∎
Theorem. (mul_div_SNo_invL)
∀x y, SNo x → SNo y → y ≠ 0 → (x :/: y) * y = x
In Proofgold the corresponding term root is 2484b5... and proposition id is 07d80b...
Proof:
The rest of this subproof is missing.
∎
Theorem. (mul_div_SNo_invR)
∀x y, SNo x → SNo y → y ≠ 0 → y * (x :/: y) = x
In Proofgold the corresponding term root is 9b260e... and proposition id is df334a...
Proof:
The rest of this subproof is missing.
∎
Theorem. (mul_div_SNo_R)
∀x y z, SNo x → SNo y → SNo z → (x :/: y) * z = (x * z) :/: y
In Proofgold the corresponding term root is a78538... and proposition id is daa8eb...
Proof:
The rest of this subproof is missing.
∎
Theorem. (mul_div_SNo_L)
∀x y z, SNo x → SNo y → SNo z → z * (x :/: y) = (z * x) :/: y
In Proofgold the corresponding term root is 95a4c6... and proposition id is c10e5c...
Proof:
The rest of this subproof is missing.
∎
Theorem. (div_mul_SNo_invL)
∀x y, SNo x → SNo y → y ≠ 0 → (x * y) :/: y = x
In Proofgold the corresponding term root is c74971... and proposition id is b99cd5...
Proof:
The rest of this subproof is missing.
∎
Theorem. (div_div_SNo)
∀x y z, SNo x → SNo y → SNo z → (x :/: y) :/: z = x :/: (y * z)
In Proofgold the corresponding term root is 3e9827... and proposition id is de0369...
Proof:
The rest of this subproof is missing.
∎
Theorem. (mul_div_SNo_both)
∀x y z w, SNo x → SNo y → SNo z → SNo w → (x :/: y) * (z :/: w) = (x * z) :/: (y * w)
In Proofgold the corresponding term root is 06b28b... and proposition id is e61f18...
Proof:
The rest of this subproof is missing.
∎
Theorem. (recip_SNo_pos_pos)
∀x, SNo x → 0 < x → 0 < recip_SNo_pos x
In Proofgold the corresponding term root is c95744... and proposition id is 30d6a5...
Proof:
The rest of this subproof is missing.
∎
Theorem. (div_SNo_pos_pos)
∀x y, SNo x → SNo y → 0 < x → 0 < y → 0 < x :/: y
In Proofgold the corresponding term root is acf73b... and proposition id is 9935dd...
Proof:
The rest of this subproof is missing.
∎
Theorem. (div_SNo_neg_pos)
∀x y, SNo x → SNo y → x < 0 → 0 < y → x :/: y < 0
In Proofgold the corresponding term root is ff8d4c... and proposition id is df8e4a...
Proof:
The rest of this subproof is missing.
∎
Theorem. (div_SNo_pos_LtL)
∀x y z, SNo x → SNo y → SNo z → 0 < y → x < z * y → x :/: y < z
In Proofgold the corresponding term root is 6ffbf3... and proposition id is 87c1ac...
Proof:
The rest of this subproof is missing.
∎
Theorem. (div_SNo_pos_LtR)
∀x y z, SNo x → SNo y → SNo z → 0 < y → z * y < x → z < x :/: y
In Proofgold the corresponding term root is caaa93... and proposition id is c36da0...
Proof:
The rest of this subproof is missing.
∎
Theorem. (div_SNo_pos_LtL')
∀x y z, SNo x → SNo y → SNo z → 0 < y → x :/: y < z → x < z * y
In Proofgold the corresponding term root is 6d81ed... and proposition id is de6389...
Proof:
The rest of this subproof is missing.
∎
Theorem. (div_SNo_pos_LtR')
∀x y z, SNo x → SNo y → SNo z → 0 < y → z < x :/: y → z * y < x
In Proofgold the corresponding term root is 0c5a47... and proposition id is a7e3d6...
Proof:
The rest of this subproof is missing.
∎
Theorem. (mul_div_SNo_nonzero_eq)
∀x y z, SNo x → SNo y → SNo z → y ≠ 0 → x = y * z → x :/: y = z
In Proofgold the corresponding term root is 4c4da2... and proposition id is bea168...
Proof:
The rest of this subproof is missing.
∎
End of Section SurrealDiv
Beginning of Section Reals
Notation. We use - as a prefix operator with priority 358 corresponding to applying term minus_SNo.
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_SNo.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_SNo.
Notation. We use :/: as an infix operator with priority 353 and no associativity corresponding to applying term div_SNo.
Notation. We use ^ as an infix operator with priority 342 and which associates to the right corresponding to applying term exp_SNo_nat.
Notation. We use < as an infix operator with priority 490 and no associativity corresponding to applying term SNoLt.
Notation. We use ≤ as an infix operator with priority 490 and no associativity corresponding to applying term SNoLe.
Theorem. (SNoS_omega_drat_intvl)
∀x ∈ SNoS_ ω, ∀k ∈ ω, ∃q ∈ SNoS_ ω, q < x ∧ x < q + eps_ k
In Proofgold the corresponding term root is d24dbb... and proposition id is eb9878...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNoS_ordsucc_omega_bdd_above)
∀x ∈ SNoS_ (ordsucc ω), x < ω → ∃N ∈ ω, x < N
In Proofgold the corresponding term root is c51d13... and proposition id is 7fd9f0...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNoS_ordsucc_omega_bdd_below)
∀x ∈ SNoS_ (ordsucc ω), - ω < x → ∃N ∈ ω, - N < x
In Proofgold the corresponding term root is d7d14e... and proposition id is 09271c...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNoS_ordsucc_omega_bdd_drat_intvl)
∀x ∈ SNoS_ (ordsucc ω), - ω < x → x < ω → ∀k ∈ ω, ∃q ∈ SNoS_ ω, q < x ∧ x < q + eps_ k
In Proofgold the corresponding term root is 57512e... and proposition id is 0ae7ab...
Proof:
The rest of this subproof is missing.
∎
Definition. We define real to be {x ∈ SNoS_ (ordsucc ω)|x ≠ ω ∧ x ≠ - ω ∧ (∀q ∈ SNoS_ ω, (∀k ∈ ω, abs_SNo (q + - x) < eps_ k) → q = x)} of type set.
In Proofgold the corresponding term root is e26ffa... and object id is 5277ef...
Theorem. (real_I)
∀x ∈ SNoS_ (ordsucc ω), x ≠ ω → x ≠ - ω → (∀q ∈ SNoS_ ω, (∀k ∈ ω, abs_SNo (q + - x) < eps_ k) → q = x) → x ∈ real
In Proofgold the corresponding term root is b05b9f... and proposition id is 9f45f4...
Proof:
The rest of this subproof is missing.
∎
Theorem. (real_E)
∀x ∈ real, ∀p : prop, (SNo x → SNoLev x ∈ ordsucc ω → x ∈ SNoS_ (ordsucc ω) → - ω < x → x < ω → (∀q ∈ SNoS_ ω, (∀k ∈ ω, abs_SNo (q + - x) < eps_ k) → q = x) → (∀k ∈ ω, ∃q ∈ SNoS_ ω, q < x ∧ x < q + eps_ k) → p) → p
In Proofgold the corresponding term root is ac0c5d... and proposition id is 447bb9...
Proof:
The rest of this subproof is missing.
∎
Theorem. (real_SNo)
∀x ∈ real, SNo x
In Proofgold the corresponding term root is 17c3d6... and proposition id is d4cbb8...
Proof:
The rest of this subproof is missing.
∎
Theorem. (real_SNoS_omega_prop)
∀x ∈ real, ∀q ∈ SNoS_ ω, (∀k ∈ ω, abs_SNo (q + - x) < eps_ k) → q = x
In Proofgold the corresponding term root is 6da66c... and proposition id is cb7f5e...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNoS_omega_real)
SNoS_ ω ⊆ real
In Proofgold the corresponding term root is a469c6... and proposition id is d9411b...
Proof:
The rest of this subproof is missing.
∎
Theorem. (real_0)
0 ∈ real
In Proofgold the corresponding term root is 53bbc2... and proposition id is 493130...
Proof:
The rest of this subproof is missing.
∎
Theorem. (real_1)
1 ∈ real
In Proofgold the corresponding term root is bbf720... and proposition id is 790b4d...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNoLev_In_real_SNoS_omega)
∀x ∈ real, ∀w, SNo w → SNoLev w ∈ SNoLev x → w ∈ SNoS_ ω
In Proofgold the corresponding term root is defc4e... and proposition id is 30df9a...
Proof:
The rest of this subproof is missing.
∎
Theorem. (real_SNoCut_SNoS_omega)
∀L R ⊆ SNoS_ ω, SNoCutP L R → L ≠ 0 → R ≠ 0 → (∀w ∈ L, ∃w' ∈ L, w < w') → (∀z ∈ R, ∃z' ∈ R, z' < z) → SNoCut L R ∈ real
In Proofgold the corresponding term root is 9f1398... and proposition id is 49872c...
Proof:
The rest of this subproof is missing.
∎
Theorem. (real_SNoCut)
∀L R ⊆ real, SNoCutP L R → L ≠ 0 → R ≠ 0 → (∀w ∈ L, ∃w' ∈ L, w < w') → (∀z ∈ R, ∃z' ∈ R, z' < z) → SNoCut L R ∈ real
In Proofgold the corresponding term root is 6c4b7d... and proposition id is 463b8d...
Proof:
The rest of this subproof is missing.
∎
Theorem. (minus_SNo_prereal_1)
∀x, SNo x → (∀q ∈ SNoS_ ω, (∀k ∈ ω, abs_SNo (q + - x) < eps_ k) → q = x) → (∀q ∈ SNoS_ ω, (∀k ∈ ω, abs_SNo (q + - - x) < eps_ k) → q = - x)
In Proofgold the corresponding term root is 824b89... and proposition id is 1d4018...
Proof:
The rest of this subproof is missing.
∎
Theorem. (minus_SNo_prereal_2)
∀x, SNo x → (∀k ∈ ω, ∃q ∈ SNoS_ ω, q < x ∧ x < q + eps_ k) → (∀k ∈ ω, ∃q ∈ SNoS_ ω, q < - x ∧ - x < q + eps_ k)
In Proofgold the corresponding term root is 68fe96... and proposition id is 8ce93f...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNo_prereal_incr_lower_pos)
∀x, SNo x → 0 < x → (∀q ∈ SNoS_ ω, (∀k ∈ ω, abs_SNo (q + - x) < eps_ k) → q = x) → (∀k ∈ ω, ∃q ∈ SNoS_ ω, q < x ∧ x < q + eps_ k) → ∀k ∈ ω, ∀p : prop, (∀q ∈ SNoS_ ω, 0 < q → q < x → x < q + eps_ k → p) → p
In Proofgold the corresponding term root is 96a462... and proposition id is 947a7e...
Proof:
The rest of this subproof is missing.
∎
Theorem. (real_minus_SNo)
∀x ∈ real, - x ∈ real
In Proofgold the corresponding term root is e8e8be... and proposition id is 4b8379...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNo_prereal_incr_lower_approx)
∀x, SNo x → (∀q ∈ SNoS_ ω, (∀k ∈ ω, abs_SNo (q + - x) < eps_ k) → q = x) → (∀k ∈ ω, ∃q ∈ SNoS_ ω, q < x ∧ x < q + eps_ k) → ∃f ∈ SNoS_ ωω, ∀n ∈ ω, f n < x ∧ x < f n + eps_ n ∧ ∀i ∈ n, f i < f n
In Proofgold the corresponding term root is 29e857... and proposition id is 85c0ef...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNo_prereal_decr_upper_approx)
∀x, SNo x → (∀q ∈ SNoS_ ω, (∀k ∈ ω, abs_SNo (q + - x) < eps_ k) → q = x) → (∀k ∈ ω, ∃q ∈ SNoS_ ω, q < x ∧ x < q + eps_ k) → ∃g ∈ SNoS_ ωω, ∀n ∈ ω, g n + - eps_ n < x ∧ x < g n ∧ ∀i ∈ n, g n < g i
In Proofgold the corresponding term root is e28afc... and proposition id is 07ed25...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNoCutP_SNoCut_lim)
∀lambda, ordinal lambda → (∀alpha ∈ lambda, ordsucc alpha ∈ lambda) → ∀L R ⊆ SNoS_ lambda, SNoCutP L R → SNoLev (SNoCut L R) ∈ ordsucc lambda
In Proofgold the corresponding term root is 1a37c9... and proposition id is f1520f...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNoCutP_SNoCut_omega)
∀L R ⊆ SNoS_ ω, SNoCutP L R → SNoLev (SNoCut L R) ∈ ordsucc ω
In Proofgold the corresponding term root is d11617... and proposition id is a23f8a...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNo_approx_real_lem)
∀f g ∈ SNoS_ ωω, (∀n m ∈ ω, f n < g m) → ∀p : prop, (SNoCutP {f n|n ∈ ω} {g n|n ∈ ω} → SNo (SNoCut {f n|n ∈ ω} {g n|n ∈ ω}) → SNoLev (SNoCut {f n|n ∈ ω} {g n|n ∈ ω}) ∈ ordsucc ω → SNoCut {f n|n ∈ ω} {g n|n ∈ ω} ∈ SNoS_ (ordsucc ω) → (∀n ∈ ω, f n < SNoCut {f n|n ∈ ω} {g n|n ∈ ω}) → (∀n ∈ ω, SNoCut {f n|n ∈ ω} {g n|n ∈ ω} < g n) → p) → p
In Proofgold the corresponding term root is d275e7... and proposition id is 3ca0b1...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNo_approx_real)
∀x, SNo x → ∀f g ∈ SNoS_ ωω, (∀n ∈ ω, f n < x) → (∀n ∈ ω, x < f n + eps_ n) → (∀n ∈ ω, ∀i ∈ n, f i < f n) → (∀n ∈ ω, x < g n) → (∀n ∈ ω, ∀i ∈ n, g n < g i) → x = SNoCut {f n|n ∈ ω} {g n|n ∈ ω} → x ∈ real
In Proofgold the corresponding term root is 8b24c5... and proposition id is fca80e...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNo_approx_real_rep)
∀x ∈ real, ∀p : prop, (∀f g ∈ SNoS_ ωω, (∀n ∈ ω, f n < x) → (∀n ∈ ω, x < f n + eps_ n) → (∀n ∈ ω, ∀i ∈ n, f i < f n) → (∀n ∈ ω, g n + - eps_ n < x) → (∀n ∈ ω, x < g n) → (∀n ∈ ω, ∀i ∈ n, g n < g i) → SNoCutP {f n|n ∈ ω} {g n|n ∈ ω} → x = SNoCut {f n|n ∈ ω} {g n|n ∈ ω} → p) → p
In Proofgold the corresponding term root is 91befd... and proposition id is a38d26...
Proof:
The rest of this subproof is missing.
∎
Theorem. (real_add_SNo)
∀x y ∈ real, x + y ∈ real
In Proofgold the corresponding term root is 695361... and proposition id is 3c841f...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNoS_ordsucc_omega_bdd_eps_pos)
∀x ∈ SNoS_ (ordsucc ω), 0 < x → x < ω → ∃N ∈ ω, eps_ N * x < 1
In Proofgold the corresponding term root is 064957... and proposition id is 9096d4...
Proof:
The rest of this subproof is missing.
∎
Theorem. (real_mul_SNo_pos)
∀x y ∈ real, 0 < x → 0 < y → x * y ∈ real
In Proofgold the corresponding term root is dfe1b6... and proposition id is 8ba463...
Proof:
The rest of this subproof is missing.
∎
Theorem. (real_mul_SNo)
∀x y ∈ real, x * y ∈ real
In Proofgold the corresponding term root is b48f84... and proposition id is 51dc42...
Proof:
The rest of this subproof is missing.
∎
Theorem. (nonneg_real_nat_interval)
∀x ∈ real, 0 ≤ x → ∃n ∈ ω, n ≤ x ∧ x < ordsucc n
In Proofgold the corresponding term root is 00f333... and proposition id is a6e1f8...
Proof:
The rest of this subproof is missing.
∎
Theorem. (pos_real_left_approx_double)
∀x ∈ real, 0 < x → x ≠ 2 → (∀m ∈ ω, x ≠ eps_ m) → ∃w ∈ SNoL_pos x, x < 2 * w
In Proofgold the corresponding term root is 721fe7... and proposition id is b80017...
Proof:
The rest of this subproof is missing.
∎
Theorem. (real_recip_SNo_lem1)
∀x, SNo x → x ∈ real → 0 < x → recip_SNo_pos x ∈ real ∧ ∀k, nat_p k → (SNo_recipaux x recip_SNo_pos k 0 ⊆ real) ∧ (SNo_recipaux x recip_SNo_pos k 1 ⊆ real)
In Proofgold the corresponding term root is 129489... and proposition id is cfc2ec...
Proof:
The rest of this subproof is missing.
∎
Theorem. (real_recip_SNo_pos)
∀x ∈ real, 0 < x → recip_SNo_pos x ∈ real
In Proofgold the corresponding term root is 8169ab... and proposition id is 406f08...
Proof:
The rest of this subproof is missing.
∎
Theorem. (real_recip_SNo)
∀x ∈ real, recip_SNo x ∈ real
In Proofgold the corresponding term root is 336956... and proposition id is 64deb9...
Proof:
The rest of this subproof is missing.
∎
Theorem. (real_div_SNo)
∀x y ∈ real, x :/: y ∈ real
In Proofgold the corresponding term root is 1b354d... and proposition id is 4ff694...
Proof:
The rest of this subproof is missing.
∎
End of Section Reals
Beginning of Section even_odd
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_nat.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_nat.
Theorem. (nat_le2_cases)
∀m, nat_p m → m ⊆ 2 → m = 0 ∨ m = 1 ∨ m = 2
In Proofgold the corresponding term root is baf753... and proposition id is 166808...
Proof:
The rest of this subproof is missing.
∎
Theorem. (prime_nat_2_lem)
∀m, nat_p m → ∀n, nat_p n → m * n = 2 → m = 1 ∨ m = 2
In Proofgold the corresponding term root is acdedc... and proposition id is 4e2fe6...
Proof:
The rest of this subproof is missing.
∎
Theorem. (prime_nat_2)
prime_nat 2
In Proofgold the corresponding term root is 3729fc... and proposition id is 9dc412...
Proof:
The rest of this subproof is missing.
∎
Notation. We use - as a prefix operator with priority 358 corresponding to applying term minus_SNo.
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_SNo.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_SNo.
Theorem. (not_eq_2m_2n1)
∀m n ∈ int, 2 * m ≠ 2 * n + 1
In Proofgold the corresponding term root is db57c3... and proposition id is af26fb...
Proof:
The rest of this subproof is missing.
∎
End of Section even_odd
Beginning of Section form100_22b
Let tag : set → set ≝ λalpha ⇒ SetAdjoin alpha {1}
Notation. We use ' as a postfix operator with priority 100 corresponding to applying term tag.
Notation. We use - as a prefix operator with priority 358 corresponding to applying term minus_SNo.
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_SNo.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_SNo.
Theorem. (atleastp_SNoS_ordsucc_omega_Power_omega)
atleastp (SNoS_ (ordsucc ω)) (𝒫 ω)
In Proofgold the corresponding term root is dcf01d... and proposition id is e3c12b...
Proof:
The rest of this subproof is missing.
∎
Theorem. (Repl_finite)
∀f : set → set, ∀X, finite X → finite {f x|x ∈ X}
In Proofgold the corresponding term root is 9e4868... and proposition id is 56bf85...
Proof:
The rest of this subproof is missing.
∎
Theorem. (infinite_bigger)
∀X ⊆ ω, infinite X → ∀m ∈ ω, ∃n ∈ X, m ∈ n
In Proofgold the corresponding term root is 1f57b0... and proposition id is c36de8...
Proof:
The rest of this subproof is missing.
∎
Theorem. (equip_real_Power_omega)
equip real (𝒫 ω)
In Proofgold the corresponding term root is be2eea... and proposition id is 6e3566...
Proof:
The rest of this subproof is missing.
∎
Theorem. (form100_22_real_uncountable_atleastp)
¬ atleastp real ω
In Proofgold the corresponding term root is 5bec50... and proposition id is e8cda3...
Proof:
The rest of this subproof is missing.
∎
Theorem. (form100_22_real_uncountable_equip)
¬ equip real ω
In Proofgold the corresponding term root is bea95a... and proposition id is 5a8d6a...
Proof:
The rest of this subproof is missing.
∎
Theorem. (form100_22_real_uncountable)
atleastp ω real ∧ ¬ equip real ω
In Proofgold the corresponding term root is 296976... and proposition id is c2bf76...
Proof:
The rest of this subproof is missing.
∎
End of Section form100_22b
Beginning of Section rational
Notation. We use - as a prefix operator with priority 358 corresponding to applying term minus_SNo.
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_SNo.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_SNo.
Notation. We use ^ as an infix operator with priority 342 and which associates to the right corresponding to applying term exp_SNo_nat.
Notation. We use :/: as an infix operator with priority 353 and no associativity corresponding to applying term div_SNo.
Notation. We use < as an infix operator with priority 490 and no associativity corresponding to applying term SNoLt.
Notation. We use ≤ as an infix operator with priority 490 and no associativity corresponding to applying term SNoLe.
Definition. We define rational to be {x ∈ real|∃m ∈ int, ∃n ∈ ω ∖ {0}, x = m :/: n} of type set.
In Proofgold the corresponding term root is efaf16... and object id is 6ea078...
End of Section rational
Beginning of Section form100_3
Notation. We use - as a prefix operator with priority 358 corresponding to applying term minus_SNo.
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_SNo.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_SNo.
Notation. We use :/: as an infix operator with priority 353 and no associativity corresponding to applying term div_SNo.
Notation. We use ^ as an infix operator with priority 342 and which associates to the right corresponding to applying term exp_SNo_nat.
Theorem. (Subq_int_SNoS_omega)
int ⊆ SNoS_ ω
In Proofgold the corresponding term root is 947247... and proposition id is 412d00...
Proof:
The rest of this subproof is missing.
∎
Theorem. (Subq_SNoS_omega_rational)
SNoS_ ω ⊆ rational
In Proofgold the corresponding term root is e4a867... and proposition id is 694b2d...
Proof:
The rest of this subproof is missing.
∎
Theorem. (Subq_rational_real)
rational ⊆ real
In Proofgold the corresponding term root is 0691ee... and proposition id is 2749c2...
Proof:
The rest of this subproof is missing.
∎
Theorem. (rational_minus_SNo)
∀q ∈ rational, - q ∈ rational
In Proofgold the corresponding term root is 31c42e... and proposition id is 6d2672...
Proof:
The rest of this subproof is missing.
∎
Definition. We define nat_pair to be λm n ⇒ 2 ^ m * (2 * n + 1) of type set → set → set.
In Proofgold the corresponding term root is 4f6fe3... and object id is 89fb54...
Theorem. (nat_pair_In_omega)
∀m n ∈ ω, nat_pair m n ∈ ω
In Proofgold the corresponding term root is 6914a5... and proposition id is d0bc54...
Proof:
The rest of this subproof is missing.
∎
Theorem. (nat_pair_0)
∀m n m' n' ∈ ω, nat_pair m n = nat_pair m' n' → m = m'
In Proofgold the corresponding term root is 26314d... and proposition id is 1af62d...
Proof:
The rest of this subproof is missing.
∎
Theorem. (nat_pair_1)
∀m n m' n' ∈ ω, nat_pair m n = nat_pair m' n' → n = n'
In Proofgold the corresponding term root is 7f1deb... and proposition id is 7929c6...
Proof:
The rest of this subproof is missing.
∎
Theorem. (form100_3)
equip ω rational
In Proofgold the corresponding term root is a25be9... and proposition id is 680b23...
Proof:
The rest of this subproof is missing.
∎
End of Section form100_3
Beginning of Section SurrealSqrt
Notation. We use - as a prefix operator with priority 358 corresponding to applying term minus_SNo.
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_SNo.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_SNo.
Notation. We use :/: as an infix operator with priority 353 and no associativity corresponding to applying term div_SNo.
Notation. We use < as an infix operator with priority 490 and no associativity corresponding to applying term SNoLt.
Notation. We use ≤ as an infix operator with priority 490 and no associativity corresponding to applying term SNoLe.
Notation. We use ^ as an infix operator with priority 342 and which associates to the right corresponding to applying term exp_SNo_nat.
Definition. We define SNoL_nonneg to be λx ⇒ {w ∈ SNoL x|0 ≤ w} of type set → set.
In Proofgold the corresponding term root is 426e35... and object id is 1ec1d7...
Theorem. (SNoL_nonneg_0)
SNoL_nonneg 0 = 0
In Proofgold the corresponding term root is ab6dce... and proposition id is 67aa58...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNoL_nonneg_1)
SNoL_nonneg 1 = 1
In Proofgold the corresponding term root is 74d4e8... and proposition id is e3fc28...
Proof:
The rest of this subproof is missing.
∎
Definition. We define SNo_sqrtauxset to be λY Z x ⇒ ⋃y ∈ Y{(x + y * z) :/: (y + z)|z ∈ Z, 0 < y + z} of type set → set → set → set.
In Proofgold the corresponding term root is 211c64... and object id is 3fc62b...
Theorem. (SNo_sqrtauxset_I)
∀Y Z x, ∀y ∈ Y, ∀z ∈ Z, 0 < y + z → (x + y * z) :/: (y + z) ∈ SNo_sqrtauxset Y Z x
In Proofgold the corresponding term root is 4b16a1... and proposition id is c09ca4...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNo_sqrtauxset_E)
∀Y Z x, ∀u ∈ SNo_sqrtauxset Y Z x, ∀p : prop, (∀y ∈ Y, ∀z ∈ Z, 0 < y + z → u = (x + y * z) :/: (y + z) → p) → p
In Proofgold the corresponding term root is 0bec9e... and proposition id is b7810c...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNo_sqrtauxset_0)
∀Z x, SNo_sqrtauxset 0 Z x = 0
In Proofgold the corresponding term root is 2d39ce... and proposition id is db93fb...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNo_sqrtauxset_0')
∀Y x, SNo_sqrtauxset Y 0 x = 0
In Proofgold the corresponding term root is b0f585... and proposition id is d57918...
Proof:
The rest of this subproof is missing.
∎
Definition. We define SNo_sqrtaux to be λx g ⇒ nat_primrec ({g w|w ∈ SNoL_nonneg x},{g z|z ∈ SNoR x}) (λk p ⇒ (p 0 ∪ SNo_sqrtauxset (p 0) (p 1) x,p 1 ∪ SNo_sqrtauxset (p 0) (p 0) x ∪ SNo_sqrtauxset (p 1) (p 1) x)) of type set → (set → set) → set → set.
In Proofgold the corresponding term root is e233df... and object id is 2ebc89...
Theorem. (SNo_sqrtaux_0)
∀x, ∀g : set → set, SNo_sqrtaux x g 0 = ({g w|w ∈ SNoL_nonneg x},{g z|z ∈ SNoR x})
In Proofgold the corresponding term root is d46ed0... and proposition id is 182e5e...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNo_sqrtaux_S)
∀x, ∀g : set → set, ∀n, nat_p n → SNo_sqrtaux x g (ordsucc n) = (SNo_sqrtaux x g n 0 ∪ SNo_sqrtauxset (SNo_sqrtaux x g n 0) (SNo_sqrtaux x g n 1) x,SNo_sqrtaux x g n 1 ∪ SNo_sqrtauxset (SNo_sqrtaux x g n 0) (SNo_sqrtaux x g n 0) x ∪ SNo_sqrtauxset (SNo_sqrtaux x g n 1) (SNo_sqrtaux x g n 1) x)
In Proofgold the corresponding term root is bb0dd8... and proposition id is 7223d6...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNo_sqrtaux_mon_lem)
∀x, ∀g : set → set, ∀m, nat_p m → ∀n, nat_p n → SNo_sqrtaux x g m 0 ⊆ SNo_sqrtaux x g (add_nat m n) 0 ∧ SNo_sqrtaux x g m 1 ⊆ SNo_sqrtaux x g (add_nat m n) 1
In Proofgold the corresponding term root is be3c0c... and proposition id is 75cbc1...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNo_sqrtaux_mon)
∀x, ∀g : set → set, ∀m, nat_p m → ∀n, nat_p n → m ⊆ n → SNo_sqrtaux x g m 0 ⊆ SNo_sqrtaux x g n 0 ∧ SNo_sqrtaux x g m 1 ⊆ SNo_sqrtaux x g n 1
In Proofgold the corresponding term root is 53a992... and proposition id is f80eb0...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNo_sqrtaux_ext)
∀x, SNo x → ∀g h : set → set, (∀x' ∈ SNoS_ (SNoLev x), g x' = h x') → ∀k, nat_p k → SNo_sqrtaux x g k = SNo_sqrtaux x h k
In Proofgold the corresponding term root is 9e623f... and proposition id is 88e0ac...
Proof:
The rest of this subproof is missing.
∎
Beginning of Section sqrt_SNo_nonneg
Let G : set → (set → set) → set ≝ λx g ⇒ SNoCut (⋃k ∈ ωSNo_sqrtaux x g k 0) (⋃k ∈ ωSNo_sqrtaux x g k 1)
Definition. We define sqrt_SNo_nonneg to be SNo_rec_i G of type set → set.
In Proofgold the corresponding term root is 71d152... and object id is 20211a...
Theorem. (sqrt_SNo_nonneg_eq)
∀x, SNo x → sqrt_SNo_nonneg x = G x sqrt_SNo_nonneg
In Proofgold the corresponding term root is 63c3e8... and proposition id is edc75b...
Proof:
The rest of this subproof is missing.
∎
Theorem. (sqrt_SNo_nonneg_prop1a)
∀x, SNo x → 0 ≤ x → (∀w ∈ SNoS_ (SNoLev x), 0 ≤ w → SNo (sqrt_SNo_nonneg w) ∧ 0 ≤ sqrt_SNo_nonneg w ∧ sqrt_SNo_nonneg w * sqrt_SNo_nonneg w = w) → ∀k, nat_p k → (∀y ∈ SNo_sqrtaux x sqrt_SNo_nonneg k 0, SNo y ∧ 0 ≤ y ∧ y * y < x) ∧ (∀y ∈ SNo_sqrtaux x sqrt_SNo_nonneg k 1, SNo y ∧ 0 ≤ y ∧ x < y * y)
In Proofgold the corresponding term root is 111bf5... and proposition id is 44bfa1...
Proof:
The rest of this subproof is missing.
∎
Theorem. (sqrt_SNo_nonneg_prop1b)
∀x, SNo x → 0 ≤ x → (∀k, nat_p k → (∀y ∈ SNo_sqrtaux x sqrt_SNo_nonneg k 0, SNo y ∧ 0 ≤ y ∧ y * y < x) ∧ (∀y ∈ SNo_sqrtaux x sqrt_SNo_nonneg k 1, SNo y ∧ 0 ≤ y ∧ x < y * y)) → SNoCutP (⋃k ∈ ωSNo_sqrtaux x sqrt_SNo_nonneg k 0) (⋃k ∈ ωSNo_sqrtaux x sqrt_SNo_nonneg k 1)
In Proofgold the corresponding term root is 79d32b... and proposition id is 3a96af...
Proof:
The rest of this subproof is missing.
∎
Theorem. (sqrt_SNo_nonneg_prop1c)
∀x, SNo x → 0 ≤ x → SNoCutP (⋃k ∈ ωSNo_sqrtaux x sqrt_SNo_nonneg k 0) (⋃k ∈ ωSNo_sqrtaux x sqrt_SNo_nonneg k 1) → (∀z ∈ (⋃k ∈ ωSNo_sqrtaux x sqrt_SNo_nonneg k 1), ∀p : prop, (SNo z → 0 ≤ z → x < z * z → p) → p) → 0 ≤ G x sqrt_SNo_nonneg
In Proofgold the corresponding term root is afcc07... and proposition id is abe562...
Proof:
The rest of this subproof is missing.
∎
Theorem. (sqrt_SNo_nonneg_prop1d)
∀x, SNo x → 0 ≤ x → (∀w ∈ SNoS_ (SNoLev x), 0 ≤ w → SNo (sqrt_SNo_nonneg w) ∧ 0 ≤ sqrt_SNo_nonneg w ∧ sqrt_SNo_nonneg w * sqrt_SNo_nonneg w = w) → SNoCutP (⋃k ∈ ωSNo_sqrtaux x sqrt_SNo_nonneg k 0) (⋃k ∈ ωSNo_sqrtaux x sqrt_SNo_nonneg k 1) → 0 ≤ G x sqrt_SNo_nonneg → G x sqrt_SNo_nonneg * G x sqrt_SNo_nonneg < x → False
In Proofgold the corresponding term root is 9308b8... and proposition id is 9979dd...
Proof:
The rest of this subproof is missing.
∎
Theorem. (sqrt_SNo_nonneg_prop1e)
∀x, SNo x → 0 ≤ x → (∀w ∈ SNoS_ (SNoLev x), 0 ≤ w → SNo (sqrt_SNo_nonneg w) ∧ 0 ≤ sqrt_SNo_nonneg w ∧ sqrt_SNo_nonneg w * sqrt_SNo_nonneg w = w) → SNoCutP (⋃k ∈ ωSNo_sqrtaux x sqrt_SNo_nonneg k 0) (⋃k ∈ ωSNo_sqrtaux x sqrt_SNo_nonneg k 1) → 0 ≤ G x sqrt_SNo_nonneg → x < G x sqrt_SNo_nonneg * G x sqrt_SNo_nonneg → False
In Proofgold the corresponding term root is bd498d... and proposition id is c07c4e...
Proof:
The rest of this subproof is missing.
∎
Theorem. (sqrt_SNo_nonneg_prop1)
∀x, SNo x → 0 ≤ x → SNo (sqrt_SNo_nonneg x) ∧ 0 ≤ sqrt_SNo_nonneg x ∧ sqrt_SNo_nonneg x * sqrt_SNo_nonneg x = x
In Proofgold the corresponding term root is db6879... and proposition id is b09b96...
Proof:
The rest of this subproof is missing.
∎
End of Section sqrt_SNo_nonneg
Theorem. (SNo_sqrtaux_0_1_prop)
∀x, SNo x → 0 ≤ x → ∀k, nat_p k → (∀y ∈ SNo_sqrtaux x sqrt_SNo_nonneg k 0, SNo y ∧ 0 ≤ y ∧ y * y < x) ∧ (∀y ∈ SNo_sqrtaux x sqrt_SNo_nonneg k 1, SNo y ∧ 0 ≤ y ∧ x < y * y)
In Proofgold the corresponding term root is 955f65... and proposition id is a2bb45...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNo_sqrtaux_0_prop)
∀x, SNo x → 0 ≤ x → ∀k, nat_p k → ∀y ∈ SNo_sqrtaux x sqrt_SNo_nonneg k 0, SNo y ∧ 0 ≤ y ∧ y * y < x
In Proofgold the corresponding term root is abda0d... and proposition id is 8db9d4...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNo_sqrtaux_1_prop)
∀x, SNo x → 0 ≤ x → ∀k, nat_p k → ∀y ∈ SNo_sqrtaux x sqrt_SNo_nonneg k 1, SNo y ∧ 0 ≤ y ∧ x < y * y
In Proofgold the corresponding term root is d59bfa... and proposition id is 95a41e...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNo_sqrt_SNo_SNoCutP)
∀x, SNo x → 0 ≤ x → SNoCutP (⋃k ∈ ωSNo_sqrtaux x sqrt_SNo_nonneg k 0) (⋃k ∈ ωSNo_sqrtaux x sqrt_SNo_nonneg k 1)
In Proofgold the corresponding term root is 33c123... and proposition id is d04ad9...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNo_sqrt_SNo_nonneg)
∀x, SNo x → 0 ≤ x → SNo (sqrt_SNo_nonneg x)
In Proofgold the corresponding term root is 016da2... and proposition id is c5c4da...
Proof:
The rest of this subproof is missing.
∎
Theorem. (sqrt_SNo_nonneg_nonneg)
∀x, SNo x → 0 ≤ x → 0 ≤ sqrt_SNo_nonneg x
In Proofgold the corresponding term root is 40638f... and proposition id is 6c1cd0...
Proof:
The rest of this subproof is missing.
∎
Theorem. (sqrt_SNo_nonneg_sqr)
∀x, SNo x → 0 ≤ x → sqrt_SNo_nonneg x * sqrt_SNo_nonneg x = x
In Proofgold the corresponding term root is c79d06... and proposition id is 15814c...
Proof:
The rest of this subproof is missing.
∎
Theorem. (sqrt_SNo_nonneg_0)
sqrt_SNo_nonneg 0 = 0
In Proofgold the corresponding term root is 0afb2a... and proposition id is c6b1a7...
Proof:
The rest of this subproof is missing.
∎
Theorem. (sqrt_SNo_nonneg_1)
sqrt_SNo_nonneg 1 = 1
In Proofgold the corresponding term root is 1801e5... and proposition id is d5a6d5...
Proof:
The rest of this subproof is missing.
∎
Theorem. (sqrt_SNo_nonneg_0inL0)
∀x, SNo x → 0 ≤ x → 0 ∈ SNoLev x → 0 ∈ SNo_sqrtaux x sqrt_SNo_nonneg 0 0
In Proofgold the corresponding term root is afb66e... and proposition id is 6bcc63...
Proof:
The rest of this subproof is missing.
∎
Theorem. (sqrt_SNo_nonneg_Lnonempty)
∀x, SNo x → 0 ≤ x → 0 ∈ SNoLev x → (⋃k ∈ ωSNo_sqrtaux x sqrt_SNo_nonneg k 0) ≠ 0
In Proofgold the corresponding term root is 1725fa... and proposition id is c1c8c8...
Proof:
The rest of this subproof is missing.
∎
Theorem. (sqrt_SNo_nonneg_Rnonempty)
∀x, SNo x → 0 ≤ x → 1 ∈ SNoLev x → (⋃k ∈ ωSNo_sqrtaux x sqrt_SNo_nonneg k 1) ≠ 0
In Proofgold the corresponding term root is b397c9... and proposition id is 04384e...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNo_sqrtauxset_real)
∀Y Z x, Y ⊆ real → Z ⊆ real → x ∈ real → SNo_sqrtauxset Y Z x ⊆ real
In Proofgold the corresponding term root is 5d5b3c... and proposition id is d040c4...
Proof:
The rest of this subproof is missing.
∎
Theorem. (SNo_sqrtauxset_real_nonneg)
∀Y Z x, Y ⊆ {w ∈ real|0 ≤ w} → Z ⊆ {z ∈ real|0 ≤ z} → x ∈ real → 0 ≤ x → SNo_sqrtauxset Y Z x ⊆ {w ∈ real|0 ≤ w}
In Proofgold the corresponding term root is 1c421c... and proposition id is 3226b5...
Proof:
The rest of this subproof is missing.
∎
Theorem. (sqrt_SNo_nonneg_SNoS_omega)
∀x ∈ SNoS_ ω, 0 ≤ x → sqrt_SNo_nonneg x ∈ real
In Proofgold the corresponding term root is 6aa091... and proposition id is 85b0b0...
Proof:
The rest of this subproof is missing.
∎
Theorem. (sqrt_SNo_nonneg_real)
∀x ∈ real, 0 ≤ x → sqrt_SNo_nonneg x ∈ real
In Proofgold the corresponding term root is 17e0d9... and proposition id is 511860...
Proof:
The rest of this subproof is missing.
∎
End of Section SurrealSqrt
Beginning of Section form100_1
Notation. We use - as a prefix operator with priority 358 corresponding to applying term minus_SNo.
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_SNo.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_SNo.
Notation. We use :/: as an infix operator with priority 353 and no associativity corresponding to applying term div_SNo.
Theorem. (divides_int_div_SNo_int)
∀m n, divides_int m n → n :/: m ∈ int
In Proofgold the corresponding term root is 33dd36... and proposition id is a3726c...
Proof:
The rest of this subproof is missing.
∎
Theorem. (form100_1_lem1)
∀m, nat_p m → ∀n, nat_p n → m * m = 2 * n * n → n = 0
In Proofgold the corresponding term root is 8d3f60... and proposition id is 9bd188...
Proof:
The rest of this subproof is missing.
∎
Theorem. (form100_1_lem2)
∀m ∈ ω, ∀n ∈ ω ∖ 1, m * m ≠ 2 * n * n
In Proofgold the corresponding term root is 1bc378... and proposition id is 1129e9...
Proof:
The rest of this subproof is missing.
∎
Notation. We use - as a prefix operator with priority 358 corresponding to applying term minus_SNo.
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_SNo.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_SNo.
Notation. We use :/: as an infix operator with priority 353 and no associativity corresponding to applying term div_SNo.
Theorem. (sqrt_2_irrational)
sqrt_SNo_nonneg 2 ∈ real ∖ rational
In Proofgold the corresponding term root is 2c9cbf... and proposition id is 44a1b3...
Proof:
The rest of this subproof is missing.
∎
End of Section form100_1
Beginning of Section Topology
Definition. We define topology_on to be λX T ⇒ T ⊆ 𝒫 X ∧ Empty ∈ T ∧ X ∈ T ∧ (∀UFam ∈ 𝒫 T, ⋃ UFam ∈ T) ∧ (∀U ∈ T, ∀V ∈ T, U ∩ V ∈ T) of type set → set → prop.
In Proofgold the corresponding term root is bb1861... and object id is 137515...
Definition. We define open_in to be λX T U ⇒ topology_on X T ∧ U ∈ T of type set → set → set → prop.
In Proofgold the corresponding term root is eedad1... and object id is 8d51ca...
Axiom. (open_in_subset) We take the following as an axiom:
∀X T U : set, open_in X T U → U ⊆ X
In Proofgold the corresponding term root is 7f87ef... and proposition id is 9ae8df...
Axiom. (topology_elem_subset) We take the following as an axiom:
∀X T U : set, topology_on X T → U ∈ T → U ⊆ X
In Proofgold the corresponding term root is 8ae492... and proposition id is 1125cb...
Axiom. (topology_has_empty) We take the following as an axiom:
∀X T : set, topology_on X T → Empty ∈ T
In Proofgold the corresponding term root is ab166b... and proposition id is 4c567a...
Axiom. (topology_has_X) We take the following as an axiom:
∀X T : set, topology_on X T → X ∈ T
In Proofgold the corresponding term root is df1876... and proposition id is e2e114...
Axiom. (topology_union_closed) We take the following as an axiom:
∀X T UFam : set, topology_on X T → UFam ⊆ T → ⋃ UFam ∈ T
In Proofgold the corresponding term root is 2314e1... and proposition id is 87e3f4...
Axiom. (topology_binintersect_closed) We take the following as an axiom:
∀X T U V : set, topology_on X T → U ∈ T → V ∈ T → U ∩ V ∈ T
In Proofgold the corresponding term root is 367ad1... and proposition id is 4e5488...
Definition. We define closed_in to be λX T C ⇒ topology_on X T ∧ (C ⊆ X ∧ ∃U ∈ T, C = X ∖ U) of type set → set → set → prop.
In Proofgold the corresponding term root is 6211a5... and object id is 1cef68...
Theorem. (closed_of_open_complement)
∀X T U : set, topology_on X T → U ∈ T → closed_in X T (X ∖ U)
In Proofgold the corresponding term root is fe3e1f... and proposition id is 3864ac...
Proof:
Let X be given.
Let T be given.
Let U be given.
Assume Htop HU.
We will prove topology_on X T ∧ (X ∖ U ⊆ X ∧ ∃V ∈ T, X ∖ U = X ∖ V).
Apply andI to the current goal.
An exact proof term for the current goal is Htop.
Apply andI to the current goal.
Apply setminus_Subq to the current goal.
We use U to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is HU.
Use reflexivity.
∎
Definition. We define finer_than to be λT' T ⇒ T ⊆ T' of type set → set → prop.
In Proofgold the corresponding term root is dc6cac... and object id is fd51eb...
Definition. We define coarser_than to be λT' T ⇒ T' ⊆ T of type set → set → prop.
In Proofgold the corresponding term root is 81c0ef... and object id is 317007...
Definition. We define discrete_topology to be λX ⇒ 𝒫 X of type set → set.
In Proofgold the corresponding term root is 2a38db... and object id is 0a6206...
Definition. We define indiscrete_topology to be λX ⇒ {Empty,X} of type set → set.
In Proofgold the corresponding term root is b81793... and object id is 7c7f62...
Definition. We define finite_complement_topology to be λX ⇒ {U ∈ 𝒫 X|finite (X ∖ U) ∨ U = Empty} of type set → set.
In Proofgold the corresponding term root is fe593d... and object id is 9ef7ff...
Definition. We define countable to be λX ⇒ atleastp X ω of type set → prop.
In Proofgold the corresponding term root is 9bb8ca... and object id is bdda90...
Theorem. (finite_countable)
∀X : set, finite X → countable X
In Proofgold the corresponding term root is be49c8... and proposition id is f5a333...
Proof:
Let X be given.
Assume Hfin.
Apply Hfin to the current goal.
Let n be given.
Assume Hpair: n ∈ ω ∧ equip X n.
We prove the intermediate claim Hn: n ∈ ω.
An exact proof term for the current goal is (andEL (n ∈ ω) (equip X n) Hpair).
We prove the intermediate claim Heq: equip X n.
An exact proof term for the current goal is (andER (n ∈ ω) (equip X n) Hpair).
We prove the intermediate claim Hn_sub: n ⊆ ω.
An exact proof term for the current goal is (omega_TransSet n Hn).
We prove the intermediate claim Hcount_n: atleastp n ω.
An exact proof term for the current goal is (Subq_atleastp n ω Hn_sub).
We prove the intermediate claim Hcount_X: atleastp X n.
An exact proof term for the current goal is (equip_atleastp X n Heq).
An exact proof term for the current goal is (atleastp_tra X n ω Hcount_X Hcount_n).
∎
Theorem. (countable_Empty)
countable Empty
In Proofgold the corresponding term root is d9321d... and proposition id is 55ec9d...
Proof:
An exact proof term for the current goal is (Subq_atleastp Empty ω (Subq_Empty ω)).
∎
Theorem. (Subq_countable)
∀X Y : set, countable Y → X ⊆ Y → countable X
In Proofgold the corresponding term root is f92860... and proposition id is 5584a3...
Proof:
Let X and Y be given.
Assume HcountY HsubXY.
We will prove atleastp X ω.
Apply atleastp_tra X Y ω to the current goal.
An exact proof term for the current goal is (Subq_atleastp X Y HsubXY).
An exact proof term for the current goal is HcountY.
∎
Axiom. (binunion_countable) We take the following as an axiom:
∀X Y : set, countable X → countable Y → countable (X ∪ Y)
In Proofgold the corresponding term root is b04050... and proposition id is 563e19...
Axiom. (Union_Power) We take the following as an axiom:
∀X Fam : set, Fam ⊆ 𝒫 X → ⋃ Fam ⊆ X
In Proofgold the corresponding term root is 8d8901... and proposition id is 0ea506...
Axiom. (binintersect_Power) We take the following as an axiom:
∀X U V : set, U ∈ 𝒫 X → V ∈ 𝒫 X → U ∩ V ∈ 𝒫 X
In Proofgold the corresponding term root is d4976b... and proposition id is 6ea6a1...
Axiom. (setminus_Power) We take the following as an axiom:
∀X U V : set, U ∈ 𝒫 X → X ∖ U ∈ 𝒫 X
In Proofgold the corresponding term root is 50f52e... and proposition id is d00a60...
Definition. We define countable_complement_topology to be λX ⇒ {U ∈ 𝒫 X|countable (X ∖ U) ∨ U = Empty} of type set → set.
In Proofgold the corresponding term root is 58878d... and object id is 921044...
Theorem. (discrete_topology_on)
∀X, topology_on X (discrete_topology X)
In Proofgold the corresponding term root is 6b73db... and proposition id is 262383...
Proof:
Let X be given.
We will prove 𝒫 X ⊆ 𝒫 X ∧ Empty ∈ 𝒫 X ∧ X ∈ 𝒫 X ∧ (∀UFam ∈ 𝒫 (𝒫 X), ⋃ UFam ∈ 𝒫 X) ∧ (∀U ∈ 𝒫 X, ∀V ∈ 𝒫 X, U ∩ V ∈ 𝒫 X).
Apply andI to the current goal.
We will prove ((𝒫 X ⊆ 𝒫 X ∧ Empty ∈ 𝒫 X) ∧ X ∈ 𝒫 X ∧ (∀UFam ∈ 𝒫 (𝒫 X), ⋃ UFam ∈ 𝒫 X)).
Apply andI to the current goal.
We will prove 𝒫 X ⊆ 𝒫 X ∧ Empty ∈ 𝒫 X ∧ X ∈ 𝒫 X.
Apply andI to the current goal.
We will prove 𝒫 X ⊆ 𝒫 X ∧ Empty ∈ 𝒫 X.
Apply andI to the current goal.
An exact proof term for the current goal is (Subq_ref (𝒫 X)).
Apply Empty_In_Power to the current goal.
Apply PowerI to the current goal.
An exact proof term for the current goal is (Subq_ref X).
We will prove ∀UFam ∈ 𝒫 (𝒫 X), ⋃ UFam ∈ 𝒫 X.
Let UFam be given.
Assume Hfam: UFam ∈ 𝒫 (𝒫 X).
Apply PowerI X (⋃ UFam) to the current goal.
Let x be given.
Assume HxUnion: x ∈ ⋃ UFam.
Apply UnionE_impred UFam x HxUnion to the current goal.
Let U be given.
Assume HUinx: x ∈ U.
Assume HUinFam: U ∈ UFam.
We prove the intermediate claim HFamSub: UFam ⊆ 𝒫 X.
An exact proof term for the current goal is (iffEL (UFam ∈ 𝒫 (𝒫 X)) (UFam ⊆ 𝒫 X) (PowerEq (𝒫 X) UFam) Hfam).
We prove the intermediate claim HUinPower: U ∈ 𝒫 X.
An exact proof term for the current goal is HFamSub U HUinFam.
We prove the intermediate claim HUsub: U ⊆ X.
An exact proof term for the current goal is (iffEL (U ∈ 𝒫 X) (U ⊆ X) (PowerEq X U) HUinPower).
An exact proof term for the current goal is (HUsub x HUinx).
We will prove ∀U ∈ 𝒫 X, ∀V ∈ 𝒫 X, U ∩ V ∈ 𝒫 X.
Let U be given.
Assume HU: U ∈ 𝒫 X.
Let V be given.
Assume HV: V ∈ 𝒫 X.
Apply PowerI X (U ∩ V) to the current goal.
Let x be given.
Assume Hxcap: x ∈ U ∩ V.
Apply binintersectE U V x Hxcap to the current goal.
Assume HxU HxV.
We prove the intermediate claim HUsub: U ⊆ X.
An exact proof term for the current goal is (iffEL (U ∈ 𝒫 X) (U ⊆ X) (PowerEq X U) HU).
An exact proof term for the current goal is (HUsub x HxU).
∎
Theorem. (indiscrete_topology_on)
∀X, topology_on X (indiscrete_topology X)
In Proofgold the corresponding term root is ecf1ef... and proposition id is 8ce7dc...
Proof:
Let X be given.
We will prove indiscrete_topology X ⊆ 𝒫 X ∧ Empty ∈ indiscrete_topology X ∧ X ∈ indiscrete_topology X ∧ (∀UFam ∈ 𝒫 (indiscrete_topology X), ⋃ UFam ∈ indiscrete_topology X) ∧ (∀U ∈ indiscrete_topology X, ∀V ∈ indiscrete_topology X, U ∩ V ∈ indiscrete_topology X).
Apply andI to the current goal.
We will prove (indiscrete_topology X ⊆ 𝒫 X ∧ Empty ∈ indiscrete_topology X) ∧ X ∈ indiscrete_topology X ∧ (∀UFam ∈ 𝒫 (indiscrete_topology X), ⋃ UFam ∈ indiscrete_topology X).
Apply andI to the current goal.
We will prove indiscrete_topology X ⊆ 𝒫 X ∧ Empty ∈ indiscrete_topology X ∧ X ∈ indiscrete_topology X.
Apply andI to the current goal.
We will prove indiscrete_topology X ⊆ 𝒫 X ∧ Empty ∈ indiscrete_topology X.
Apply andI to the current goal.
Let U be given.
Assume HU: U ∈ indiscrete_topology X.
Apply UPairE U Empty X HU to the current goal.
Assume HUe: U = Empty.
rewrite the current goal using HUe (from left to right).
An exact proof term for the current goal is (Empty_In_Power X).
Assume HUX: U = X.
rewrite the current goal using HUX (from left to right).
An exact proof term for the current goal is (Self_In_Power X).
An exact proof term for the current goal is (UPairI1 Empty X).
An exact proof term for the current goal is (UPairI2 Empty X).
We will prove ∀UFam ∈ 𝒫 (indiscrete_topology X), ⋃ UFam ∈ indiscrete_topology X.
Let UFam be given.
Assume Hfam: UFam ∈ 𝒫 (indiscrete_topology X).
We prove the intermediate claim Hsub: UFam ⊆ indiscrete_topology X.
An exact proof term for the current goal is (PowerE (indiscrete_topology X) UFam Hfam).
Apply xm (∃U : set, U ∈ UFam ∧ U = X) to the current goal.
Assume Hex: ∃U : set, U ∈ UFam ∧ U = X.
We prove the intermediate claim HUnion_sub: ⋃ UFam ⊆ X.
Let x be given.
Assume HxUnion.
Apply UnionE_impred UFam x HxUnion to the current goal.
Let U be given.
Assume HxU HUin.
We prove the intermediate claim HUtop: U ∈ indiscrete_topology X.
An exact proof term for the current goal is (Hsub U HUin).
Apply UPairE U Empty X HUtop to the current goal.
Assume HUe: U = Empty.
We prove the intermediate claim HxEmpty: x ∈ Empty.
rewrite the current goal using HUe (from right to left).
An exact proof term for the current goal is HxU.
An exact proof term for the current goal is (EmptyE x HxEmpty (x ∈ X)).
Assume HUX: U = X.
rewrite the current goal using HUX (from right to left).
An exact proof term for the current goal is HxU.
We prove the intermediate claim HX_sub: X ⊆ ⋃ UFam.
Let x be given.
Assume HxX.
Apply Hex to the current goal.
Let U be given.
Assume HUinpair: U ∈ UFam ∧ U = X.
We prove the intermediate claim HUin: U ∈ UFam.
An exact proof term for the current goal is (andEL (U ∈ UFam) (U = X) HUinpair).
We prove the intermediate claim HUeq: U = X.
An exact proof term for the current goal is (andER (U ∈ UFam) (U = X) HUinpair).
We prove the intermediate claim HxU: x ∈ U.
rewrite the current goal using HUeq (from left to right).
An exact proof term for the current goal is HxX.
Apply UnionI UFam x U HxU HUin to the current goal.
We prove the intermediate claim HUnion_eq: ⋃ UFam = X.
Apply set_ext to the current goal.
An exact proof term for the current goal is HUnion_sub.
An exact proof term for the current goal is HX_sub.
rewrite the current goal using HUnion_eq (from left to right).
An exact proof term for the current goal is (UPairI2 Empty X).
Assume Hnone: ¬ ∃U : set, U ∈ UFam ∧ U = X.
We prove the intermediate claim HUnion_empty: ⋃ UFam = Empty.
Apply Empty_Subq_eq to the current goal.
Let x be given.
Assume HxUnion.
Apply UnionE_impred UFam x HxUnion to the current goal.
Let U be given.
Assume HxU HUin.
We prove the intermediate claim HUtop: U ∈ indiscrete_topology X.
An exact proof term for the current goal is (Hsub U HUin).
Apply UPairE U Empty X HUtop to the current goal.
Assume HUe: U = Empty.
We prove the intermediate claim HxEmpty: x ∈ Empty.
rewrite the current goal using HUe (from right to left).
An exact proof term for the current goal is HxU.
An exact proof term for the current goal is HxEmpty.
Assume HUX: U = X.
Apply FalseE to the current goal.
Apply Hnone to the current goal.
We use U to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is HUin.
An exact proof term for the current goal is HUX.
rewrite the current goal using HUnion_empty (from left to right).
An exact proof term for the current goal is (UPairI1 Empty X).
We will prove ∀U ∈ indiscrete_topology X, ∀V ∈ indiscrete_topology X, U ∩ V ∈ indiscrete_topology X.
Let U be given.
Assume HU: U ∈ indiscrete_topology X.
Let V be given.
Assume HV: V ∈ indiscrete_topology X.
Apply UPairE U Empty X HU to the current goal.
Assume HUe: U = Empty.
We prove the intermediate claim Hcap: U ∩ V = Empty.
rewrite the current goal using HUe (from left to right).
Apply Empty_Subq_eq to the current goal.
An exact proof term for the current goal is (binintersect_Subq_1 Empty V).
rewrite the current goal using Hcap (from left to right).
An exact proof term for the current goal is (UPairI1 Empty X).
Assume HUX: U = X.
Apply UPairE V Empty X HV to the current goal.
Assume HVe: V = Empty.
We prove the intermediate claim Hcap: U ∩ V = Empty.
rewrite the current goal using HVe (from left to right).
Apply Empty_Subq_eq to the current goal.
An exact proof term for the current goal is (binintersect_Subq_2 U Empty).
rewrite the current goal using Hcap (from left to right).
An exact proof term for the current goal is (UPairI1 Empty X).
Assume HVX: V = X.
We prove the intermediate claim Hcap: U ∩ V = X.
Apply set_ext to the current goal.
rewrite the current goal using HUX (from left to right).
rewrite the current goal using HVX (from left to right).
An exact proof term for the current goal is (binintersect_Subq_1 X X).
Let x be given.
Assume HxX.
rewrite the current goal using HUX (from left to right).
rewrite the current goal using HVX (from left to right).
An exact proof term for the current goal is (binintersectI X X x HxX HxX).
rewrite the current goal using Hcap (from left to right).
An exact proof term for the current goal is (UPairI2 Empty X).
∎
Theorem. (finite_complement_topology_on)
∀X, topology_on X (finite_complement_topology X)
In Proofgold the corresponding term root is 6f0b3b... and proposition id is a98a95...
Proof:
Let X be given.
We prove the intermediate claim HEmptyOpen: Empty ∈ finite_complement_topology X.
An exact proof term for the current goal is (SepI (𝒫 X) (λU0 : set ⇒ finite (X ∖ U0) ∨ U0 = Empty) Empty (Empty_In_Power X) (orIR (finite (X ∖ Empty)) (Empty = Empty) (λP H ⇒ H))).
We will prove finite_complement_topology X ⊆ 𝒫 X ∧ Empty ∈ finite_complement_topology X ∧ X ∈ finite_complement_topology X ∧ (∀UFam ∈ 𝒫 (finite_complement_topology X), ⋃ UFam ∈ finite_complement_topology X) ∧ (∀U ∈ finite_complement_topology X, ∀V ∈ finite_complement_topology X, U ∩ V ∈ finite_complement_topology X).
Apply andI to the current goal.
We will prove (finite_complement_topology X ⊆ 𝒫 X ∧ Empty ∈ finite_complement_topology X) ∧ X ∈ finite_complement_topology X ∧ (∀UFam ∈ 𝒫 (finite_complement_topology X), ⋃ UFam ∈ finite_complement_topology X).
Apply andI to the current goal.
We will prove finite_complement_topology X ⊆ 𝒫 X ∧ Empty ∈ finite_complement_topology X ∧ X ∈ finite_complement_topology X.
Apply andI to the current goal.
We will prove finite_complement_topology X ⊆ 𝒫 X ∧ Empty ∈ finite_complement_topology X.
Apply andI to the current goal.
Let U be given.
Assume HU: U ∈ finite_complement_topology X.
An exact proof term for the current goal is (SepE1 (𝒫 X) (λU0 : set ⇒ finite (X ∖ U0) ∨ U0 = Empty) U HU).
An exact proof term for the current goal is (SepI (𝒫 X) (λU0 : set ⇒ finite (X ∖ U0) ∨ U0 = Empty) Empty (Empty_In_Power X) (orIR (finite (X ∖ Empty)) (Empty = Empty) (λP H ⇒ H))).
We prove the intermediate claim Hdiff_empty: X ∖ X = Empty.
Apply Empty_Subq_eq to the current goal.
Let x be given.
Assume Hx.
We prove the intermediate claim HxX: x ∈ X.
An exact proof term for the current goal is (setminusE1 X X x Hx).
We prove the intermediate claim Hxnot: x ∉ X.
An exact proof term for the current goal is (setminusE2 X X x Hx).
Apply FalseE to the current goal.
An exact proof term for the current goal is (Hxnot HxX).
We prove the intermediate claim HfinDiff: finite (X ∖ X).
rewrite the current goal using Hdiff_empty (from left to right).
An exact proof term for the current goal is finite_Empty.
An exact proof term for the current goal is (SepI (𝒫 X) (λU0 : set ⇒ finite (X ∖ U0) ∨ U0 = Empty) X (Self_In_Power X) (orIL (finite (X ∖ X)) (X = Empty) HfinDiff)).
We will prove ∀UFam ∈ 𝒫 (finite_complement_topology X), ⋃ UFam ∈ finite_complement_topology X.
Let UFam be given.
Assume Hfam: UFam ∈ 𝒫 (finite_complement_topology X).
We prove the intermediate claim Hsub: UFam ⊆ finite_complement_topology X.
An exact proof term for the current goal is (PowerE (finite_complement_topology X) UFam Hfam).
Apply xm (∃U : set, U ∈ UFam ∧ finite (X ∖ U)) to the current goal.
Assume Hex: ∃U : set, U ∈ UFam ∧ finite (X ∖ U).
We prove the intermediate claim HUnionInPower: ⋃ UFam ∈ 𝒫 X.
Apply PowerI X (⋃ UFam) to the current goal.
Let x be given.
Assume HxUnion.
Apply UnionE_impred UFam x HxUnion to the current goal.
Let U be given.
Assume HxU HUin.
We prove the intermediate claim HUinPow: U ∈ 𝒫 X.
An exact proof term for the current goal is (SepE1 (𝒫 X) (λU0 : set ⇒ finite (X ∖ U0) ∨ U0 = Empty) U (Hsub U HUin)).
We prove the intermediate claim HUsub: U ⊆ X.
An exact proof term for the current goal is (PowerE X U HUinPow).
An exact proof term for the current goal is (HUsub x HxU).
We prove the intermediate claim HUnionPred: finite (X ∖ ⋃ UFam) ∨ ⋃ UFam = Empty.
Apply orIL to the current goal.
Apply Hex to the current goal.
Let U be given.
Assume Hpair: U ∈ UFam ∧ finite (X ∖ U).
We prove the intermediate claim HUin: U ∈ UFam.
An exact proof term for the current goal is (andEL (U ∈ UFam) (finite (X ∖ U)) Hpair).
We prove the intermediate claim HUfin: finite (X ∖ U).
An exact proof term for the current goal is (andER (U ∈ UFam) (finite (X ∖ U)) Hpair).
We prove the intermediate claim Hsubset: X ∖ ⋃ UFam ⊆ X ∖ U.
Let x be given.
Assume Hx.
We prove the intermediate claim HxX: x ∈ X.
An exact proof term for the current goal is (setminusE1 X (⋃ UFam) x Hx).
We prove the intermediate claim HnotUnion: x ∉ ⋃ UFam.
An exact proof term for the current goal is (setminusE2 X (⋃ UFam) x Hx).
We prove the intermediate claim HnotU: x ∉ U.
Assume HxU.
Apply HnotUnion to the current goal.
Apply UnionI UFam x U HxU HUin to the current goal.
Apply setminusI X U x HxX HnotU to the current goal.
An exact proof term for the current goal is (Subq_finite (X ∖ U) HUfin (X ∖ ⋃ UFam) Hsubset).
An exact proof term for the current goal is (SepI (𝒫 X) (λU0 : set ⇒ finite (X ∖ U0) ∨ U0 = Empty) (⋃ UFam) HUnionInPower HUnionPred).
Assume Hnone: ¬ ∃U : set, U ∈ UFam ∧ finite (X ∖ U).
We prove the intermediate claim HUnionEmpty: ⋃ UFam = Empty.
Apply Empty_Subq_eq to the current goal.
Let x be given.
Assume HxUnion.
Apply UnionE_impred UFam x HxUnion to the current goal.
Let U be given.
Assume HxU HUin.
We prove the intermediate claim HUdata: finite (X ∖ U) ∨ U = Empty.
An exact proof term for the current goal is (SepE2 (𝒫 X) (λU0 : set ⇒ finite (X ∖ U0) ∨ U0 = Empty) U (Hsub U HUin)).
Apply HUdata (x ∈ Empty) to the current goal.
Assume HUfin.
Apply FalseE to the current goal.
Apply Hnone to the current goal.
We use U to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is HUin.
An exact proof term for the current goal is HUfin.
Assume HUempty: U = Empty.
rewrite the current goal using HUempty (from right to left).
An exact proof term for the current goal is HxU.
rewrite the current goal using HUnionEmpty (from left to right).
An exact proof term for the current goal is HEmptyOpen.
We will prove ∀U ∈ finite_complement_topology X, ∀V ∈ finite_complement_topology X, U ∩ V ∈ finite_complement_topology X.
Let U be given.
Assume HU: U ∈ finite_complement_topology X.
Let V be given.
Assume HV: V ∈ finite_complement_topology X.
We prove the intermediate claim HUdata: finite (X ∖ U) ∨ U = Empty.
An exact proof term for the current goal is (SepE2 (𝒫 X) (λU0 : set ⇒ finite (X ∖ U0) ∨ U0 = Empty) U HU).
We prove the intermediate claim HVdata: finite (X ∖ V) ∨ V = Empty.
An exact proof term for the current goal is (SepE2 (𝒫 X) (λU0 : set ⇒ finite (X ∖ U0) ∨ U0 = Empty) V HV).
Apply HUdata (U ∩ V ∈ finite_complement_topology X) to the current goal.
Assume HUfin.
Apply HVdata (U ∩ V ∈ finite_complement_topology X) to the current goal.
Assume HVfin.
We prove the intermediate claim HcapInPower: U ∩ V ∈ 𝒫 X.
We prove the intermediate claim HUsub: U ⊆ X.
An exact proof term for the current goal is (PowerE X U (SepE1 (𝒫 X) (λU0 : set ⇒ finite (X ∖ U0) ∨ U0 = Empty) U HU)).
Apply PowerI X (U ∩ V) to the current goal.
Let x be given.
Assume HxCap.
Apply binintersectE U V x HxCap to the current goal.
Assume HxU HxV.
An exact proof term for the current goal is (HUsub x HxU).
We prove the intermediate claim HcapPred: finite (X ∖ (U ∩ V)) ∨ U ∩ V = Empty.
Apply orIL to the current goal.
We prove the intermediate claim HfinUnion: finite ((X ∖ U) ∪ (X ∖ V)).
An exact proof term for the current goal is (binunion_finite (X ∖ U) HUfin (X ∖ V) HVfin).
We prove the intermediate claim Hsubset: X ∖ (U ∩ V) ⊆ (X ∖ U) ∪ (X ∖ V).
Let x be given.
Assume Hx.
We prove the intermediate claim HxX: x ∈ X.
An exact proof term for the current goal is (setminusE1 X (U ∩ V) x Hx).
We prove the intermediate claim HnotCap: x ∉ U ∩ V.
An exact proof term for the current goal is (setminusE2 X (U ∩ V) x Hx).
Apply xm (x ∈ U) to the current goal.
Assume HxU.
We prove the intermediate claim HnotV: x ∉ V.
Assume HxV.
Apply HnotCap to the current goal.
An exact proof term for the current goal is (binintersectI U V x HxU HxV).
Apply binunionI2 (X ∖ U) (X ∖ V) to the current goal.
Apply setminusI X V x HxX HnotV to the current goal.
Assume HnotU.
Apply binunionI1 (X ∖ U) (X ∖ V) to the current goal.
Apply setminusI X U x HxX HnotU to the current goal.
An exact proof term for the current goal is (Subq_finite ((X ∖ U) ∪ (X ∖ V)) HfinUnion (X ∖ (U ∩ V)) Hsubset).
An exact proof term for the current goal is (SepI (𝒫 X) (λU0 : set ⇒ finite (X ∖ U0) ∨ U0 = Empty) (U ∩ V) HcapInPower HcapPred).
Assume HVempty: V = Empty.
We prove the intermediate claim Hcap_empty: U ∩ V = Empty.
rewrite the current goal using HVempty (from left to right).
Apply Empty_Subq_eq to the current goal.
An exact proof term for the current goal is (binintersect_Subq_2 U Empty).
rewrite the current goal using Hcap_empty (from left to right).
An exact proof term for the current goal is HEmptyOpen.
Assume HUempty: U = Empty.
We prove the intermediate claim Hcap_empty: U ∩ V = Empty.
rewrite the current goal using HUempty (from left to right).
Apply Empty_Subq_eq to the current goal.
An exact proof term for the current goal is (binintersect_Subq_1 Empty V).
rewrite the current goal using Hcap_empty (from left to right).
An exact proof term for the current goal is HEmptyOpen.
∎
Theorem. (finer_than_refl)
∀T : set, finer_than T T
In Proofgold the corresponding term root is 0f302a... and proposition id is afdc02...
Proof:
Let T be given.
An exact proof term for the current goal is (Subq_ref T).
∎
Theorem. (finer_than_trans)
∀A B C : set, finer_than B A → finer_than C B → finer_than C A
In Proofgold the corresponding term root is f1260a... and proposition id is 2f9d8f...
Proof:
Let A, B and C be given.
Assume H1: finer_than B A.
Assume H2: finer_than C B.
An exact proof term for the current goal is (Subq_tra A B C H1 H2).
∎
Theorem. (finer_coarser_dual)
∀T T' : set, finer_than T' T → coarser_than T T'
In Proofgold the corresponding term root is ddfa5b... and proposition id is 7592c5...
Proof:
Let T and T' be given.
Assume H.
An exact proof term for the current goal is H.
∎
Definition. We define comparable_topologies to be λT1 T2 ⇒ finer_than T1 T2 ∨ finer_than T2 T1 of type set → set → prop.
In Proofgold the corresponding term root is 310f6a... and object id is 209d03...
Definition. We define topology_eq to be λX T1 T2 ⇒ topology_on X T1 ∧ topology_on X T2 ∧ T1 = T2 of type set → set → set → prop.
In Proofgold the corresponding term root is 2e5f76... and object id is 6221c3...
Theorem. (topology_eq_sym)
∀X T1 T2 : set, topology_eq X T1 T2 → topology_eq X T2 T1
In Proofgold the corresponding term root is fbcc7e... and proposition id is c77487...
Proof:
Let X, T1 and T2 be given.
Assume H.
We prove the intermediate claim Hpair: topology_on X T1 ∧ topology_on X T2.
An exact proof term for the current goal is (andEL (topology_on X T1 ∧ topology_on X T2) (T1 = T2) H).
We prove the intermediate claim Heq: T1 = T2.
An exact proof term for the current goal is (andER (topology_on X T1 ∧ topology_on X T2) (T1 = T2) H).
We prove the intermediate claim HT1: topology_on X T1.
An exact proof term for the current goal is (andEL (topology_on X T1) (topology_on X T2) Hpair).
We prove the intermediate claim HT2: topology_on X T2.
An exact proof term for the current goal is (andER (topology_on X T1) (topology_on X T2) Hpair).
We will prove topology_on X T2 ∧ topology_on X T1 ∧ T2 = T1.
Apply andI to the current goal.
Apply andI to the current goal.
An exact proof term for the current goal is HT2.
An exact proof term for the current goal is HT1.
rewrite the current goal using Heq (from right to left).
Use reflexivity.
∎
Theorem. (topology_eq_trans)
∀X T1 T2 T3 : set, topology_eq X T1 T2 → topology_eq X T2 T3 → topology_eq X T1 T3
In Proofgold the corresponding term root is 32f299... and proposition id is 0f3a7d...
Proof:
Let X, T1, T2 and T3 be given.
Assume H12 H23.
We prove the intermediate claim H12pair: topology_on X T1 ∧ topology_on X T2.
An exact proof term for the current goal is (andEL (topology_on X T1 ∧ topology_on X T2) (T1 = T2) H12).
We prove the intermediate claim H12eq: T1 = T2.
An exact proof term for the current goal is (andER (topology_on X T1 ∧ topology_on X T2) (T1 = T2) H12).
We prove the intermediate claim HT1: topology_on X T1.
An exact proof term for the current goal is (andEL (topology_on X T1) (topology_on X T2) H12pair).
We prove the intermediate claim HT2: topology_on X T2.
An exact proof term for the current goal is (andER (topology_on X T1) (topology_on X T2) H12pair).
We prove the intermediate claim H23pair: topology_on X T2 ∧ topology_on X T3.
An exact proof term for the current goal is (andEL (topology_on X T2 ∧ topology_on X T3) (T2 = T3) H23).
We prove the intermediate claim H23eq: T2 = T3.
An exact proof term for the current goal is (andER (topology_on X T2 ∧ topology_on X T3) (T2 = T3) H23).
We prove the intermediate claim HT3: topology_on X T3.
An exact proof term for the current goal is (andER (topology_on X T2) (topology_on X T3) H23pair).
We will prove topology_on X T1 ∧ topology_on X T3 ∧ T1 = T3.
Apply andI to the current goal.
Apply andI to the current goal.
An exact proof term for the current goal is HT1.
An exact proof term for the current goal is HT3.
rewrite the current goal using H12eq (from left to right).
rewrite the current goal using H23eq (from left to right).
Use reflexivity.
∎
Theorem. (topology_eq_refl)
∀X T : set, topology_on X T → topology_eq X T T
In Proofgold the corresponding term root is 4dc43c... and proposition id is 671873...
Proof:
Let X and T be given.
Assume HT.
We will prove topology_on X T ∧ topology_on X T ∧ T = T.
Apply andI to the current goal.
Apply andI to the current goal.
An exact proof term for the current goal is HT.
An exact proof term for the current goal is HT.
Use reflexivity.
∎
Definition. We define strictly_finer_than to be λT' T ⇒ finer_than T' T ∧ ¬ finer_than T T' of type set → set → prop.
In Proofgold the corresponding term root is 391972... and object id is f17d93...
Definition. We define strictly_coarser_than to be λT' T ⇒ coarser_than T' T ∧ ¬ coarser_than T T' of type set → set → prop.
In Proofgold the corresponding term root is a159db... and object id is 351212...
Definition. We define discrete_topology_alt to be discrete_topology of type set → set.
In Proofgold the corresponding term root is 2a38db... and object id is 0a6206...
Definition. We define trivial_topology to be indiscrete_topology of type set → set.
In Proofgold the corresponding term root is b81793... and object id is 7c7f62...
Definition. We define finer_than_topology to be λX T' T ⇒ topology_on X T' ∧ topology_on X T ∧ finer_than T' T of type set → set → set → prop.
In Proofgold the corresponding term root is 3d83ff... and object id is 96fc54...
Theorem. (finer_than_def)
∀T T' : set, finer_than T' T ↔ coarser_than T T'
In Proofgold the corresponding term root is c0f0c4... and proposition id is 8003f0...
Proof:
Let T and T' be given.
Apply iffI to the current goal.
Assume H.
An exact proof term for the current goal is H.
Assume H.
An exact proof term for the current goal is H.
∎
Theorem. (discrete_topology_finest)
∀X T : set, topology_on X T → finer_than (discrete_topology X) T
In Proofgold the corresponding term root is 26b184... and proposition id is ba4c69...
Proof:
Let X and T be given.
Assume HT.
We prove the intermediate claim H1: ((T ⊆ 𝒫 X ∧ Empty ∈ T) ∧ X ∈ T) ∧ (∀UFam ∈ 𝒫 T, ⋃ UFam ∈ T).
An exact proof term for the current goal is (andEL (((T ⊆ 𝒫 X ∧ Empty ∈ T) ∧ X ∈ T) ∧ (∀UFam ∈ 𝒫 T, ⋃ UFam ∈ T)) (∀U ∈ T, ∀V ∈ T, U ∩ V ∈ T) HT).
We prove the intermediate claim H2: (T ⊆ 𝒫 X ∧ Empty ∈ T) ∧ X ∈ T.
An exact proof term for the current goal is (andEL ((T ⊆ 𝒫 X ∧ Empty ∈ T) ∧ X ∈ T) (∀UFam ∈ 𝒫 T, ⋃ UFam ∈ T) H1).
We prove the intermediate claim H3: T ⊆ 𝒫 X ∧ Empty ∈ T.
An exact proof term for the current goal is (andEL (T ⊆ 𝒫 X ∧ Empty ∈ T) (X ∈ T) H2).
We prove the intermediate claim HTsub: T ⊆ 𝒫 X.
An exact proof term for the current goal is (andEL (T ⊆ 𝒫 X) (Empty ∈ T) H3).
An exact proof term for the current goal is HTsub.
∎
Theorem. (indiscrete_topology_coarsest)
∀X T : set, topology_on X T → coarser_than (indiscrete_topology X) T
In Proofgold the corresponding term root is b1e55a... and proposition id is c5eae3...
Proof:
Let X and T be given.
Assume HT.
We prove the intermediate claim Hchunk1: ((T ⊆ 𝒫 X ∧ Empty ∈ T) ∧ X ∈ T) ∧ (∀UFam ∈ 𝒫 T, ⋃ UFam ∈ T).
An exact proof term for the current goal is (andEL (((T ⊆ 𝒫 X ∧ Empty ∈ T) ∧ X ∈ T) ∧ (∀UFam ∈ 𝒫 T, ⋃ UFam ∈ T)) (∀U ∈ T, ∀V ∈ T, U ∩ V ∈ T) HT).
We prove the intermediate claim Hchunk2: (T ⊆ 𝒫 X ∧ Empty ∈ T) ∧ X ∈ T.
An exact proof term for the current goal is (andEL ((T ⊆ 𝒫 X ∧ Empty ∈ T) ∧ X ∈ T) (∀UFam ∈ 𝒫 T, ⋃ UFam ∈ T) Hchunk1).
We prove the intermediate claim Hchunk3: T ⊆ 𝒫 X ∧ Empty ∈ T.
An exact proof term for the current goal is (andEL (T ⊆ 𝒫 X ∧ Empty ∈ T) (X ∈ T) Hchunk2).
We prove the intermediate claim Hempty: Empty ∈ T.
An exact proof term for the current goal is (andER (T ⊆ 𝒫 X) (Empty ∈ T) Hchunk3).
We prove the intermediate claim HX: X ∈ T.
An exact proof term for the current goal is (andER ((T ⊆ 𝒫 X) ∧ Empty ∈ T) (X ∈ T) Hchunk2).
Let U be given.
Assume HU: U ∈ indiscrete_topology X.
Apply UPairE U Empty X HU to the current goal.
Assume HUempty: U = Empty.
rewrite the current goal using HUempty (from left to right).
An exact proof term for the current goal is Hempty.
Assume HUX: U = X.
rewrite the current goal using HUX (from left to right).
An exact proof term for the current goal is HX.
∎
Theorem. (discrete_open_all)
∀X U : set, U ⊆ X → U ∈ discrete_topology X
In Proofgold the corresponding term root is 7143d7... and proposition id is 3c48d3...
Proof:
Let X and U be given.
Assume HUsub.
Apply PowerI X U HUsub to the current goal.
∎
Theorem. (indiscrete_open_iff)
∀X U : set, U ∈ indiscrete_topology X ↔ (U = Empty ∨ U = X)
In Proofgold the corresponding term root is a58418... and proposition id is 1d6de6...
Proof:
Let X and U be given.
Apply iffI to the current goal.
Assume HU.
An exact proof term for the current goal is (UPairE U Empty X HU).
Assume Hcases: U = Empty ∨ U = X.
We prove the intermediate claim HUempty_branch: U = Empty → U ∈ indiscrete_topology X.
Assume HUE: U = Empty.
rewrite the current goal using HUE (from left to right).
An exact proof term for the current goal is (UPairI1 Empty X).
We prove the intermediate claim HUx_branch: U = X → U ∈ indiscrete_topology X.
Assume HUX: U = X.
rewrite the current goal using HUX (from left to right).
An exact proof term for the current goal is (UPairI2 Empty X).
An exact proof term for the current goal is (Hcases (U ∈ indiscrete_topology X) HUempty_branch HUx_branch).
∎
Theorem. (finite_complement_topology_open_criterion)
∀X U : set, open_in X (finite_complement_topology X) U → finite (X ∖ U) ∨ U = Empty
In Proofgold the corresponding term root is 76de7d... and proposition id is 66e844...
Proof:
Let X and U be given.
Assume Hopen.
We prove the intermediate claim HUin: U ∈ finite_complement_topology X.
An exact proof term for the current goal is (andER (topology_on X (finite_complement_topology X)) (U ∈ finite_complement_topology X) Hopen).
An exact proof term for the current goal is (SepE2 (𝒫 X) (λU0 : set ⇒ finite (X ∖ U0) ∨ U0 = Empty) U HUin).
∎
Theorem. (finite_complement_topology_contains_empty)
∀X : set, Empty ∈ finite_complement_topology X
In Proofgold the corresponding term root is cf3756... and proposition id is 19683c...
Proof:
Let X be given.
An exact proof term for the current goal is (SepI (𝒫 X) (λU0 : set ⇒ finite (X ∖ U0) ∨ U0 = Empty) Empty (Empty_In_Power X) (orIR (finite (X ∖ Empty)) (Empty = Empty) (λP H ⇒ H))).
∎
Theorem. (finite_complement_topology_contains_full)
∀X : set, X ∈ finite_complement_topology X
In Proofgold the corresponding term root is a1ea79... and proposition id is 7ab81b...
Proof:
Let X be given.
We prove the intermediate claim Hdiff_empty: X ∖ X = Empty.
Apply Empty_Subq_eq to the current goal.
Let x be given.
Assume Hx.
We prove the intermediate claim Hxin: x ∈ X.
An exact proof term for the current goal is (setminusE1 X X x Hx).
We prove the intermediate claim Hxnot: x ∉ X.
An exact proof term for the current goal is (setminusE2 X X x Hx).
Apply FalseE to the current goal.
An exact proof term for the current goal is (Hxnot Hxin).
We prove the intermediate claim HfinDiff: finite (X ∖ X).
rewrite the current goal using Hdiff_empty (from left to right).
An exact proof term for the current goal is finite_Empty.
An exact proof term for the current goal is (SepI (𝒫 X) (λU0 : set ⇒ finite (X ∖ U0) ∨ U0 = Empty) X (Self_In_Power X) (orIL (finite (X ∖ X)) (X = Empty) HfinDiff)).
∎
Theorem. (countable_complement_topology_open_iff)
∀X U : set, open_in X (countable_complement_topology X) U → countable (X ∖ U) ∨ U = Empty
In Proofgold the corresponding term root is 75a7c2... and proposition id is c14c86...
Proof:
Let X and U be given.
Assume Hopen.
We prove the intermediate claim HUin: U ∈ countable_complement_topology X.
An exact proof term for the current goal is (andER (topology_on X (countable_complement_topology X)) (U ∈ countable_complement_topology X) Hopen).
An exact proof term for the current goal is (SepE2 (𝒫 X) (λU0 : set ⇒ countable (X ∖ U0) ∨ U0 = Empty) U HUin).
∎
Theorem. (countable_complement_topology_contains_empty)
∀X : set, Empty ∈ countable_complement_topology X
In Proofgold the corresponding term root is 0ee93f... and proposition id is edeb90...
Proof:
Let X be given.
An exact proof term for the current goal is (SepI (𝒫 X) (λU0 : set ⇒ countable (X ∖ U0) ∨ U0 = Empty) Empty (Empty_In_Power X) (orIR (countable (X ∖ Empty)) (Empty = Empty) (λP H ⇒ H))).
∎
Theorem. (countable_complement_topology_contains_full)
∀X : set, X ∈ countable_complement_topology X
In Proofgold the corresponding term root is 5588a4... and proposition id is 91e743...
Proof:
Let X be given.
We prove the intermediate claim Hdiff_empty: X ∖ X = Empty.
Apply Empty_Subq_eq to the current goal.
Let x be given.
Assume Hx.
We prove the intermediate claim HxX: x ∈ X.
An exact proof term for the current goal is (setminusE1 X X x Hx).
We prove the intermediate claim Hxnot: x ∉ X.
An exact proof term for the current goal is (setminusE2 X X x Hx).
Apply FalseE to the current goal.
An exact proof term for the current goal is (Hxnot HxX).
We prove the intermediate claim HcountDiff: countable (X ∖ X).
rewrite the current goal using Hdiff_empty (from left to right).
An exact proof term for the current goal is (Subq_atleastp Empty ω (Subq_Empty ω)).
An exact proof term for the current goal is (SepI (𝒫 X) (λU0 : set ⇒ countable (X ∖ U0) ∨ U0 = Empty) X (Self_In_Power X) (orIL (countable (X ∖ X)) (X = Empty) HcountDiff)).
∎
Theorem. (countable_complement_finer_than_finite_complement)
∀X : set, finer_than (countable_complement_topology X) (finite_complement_topology X)
In Proofgold the corresponding term root is 894d65... and proposition id is 7f6600...
Proof:
Let X be given.
We will prove finite_complement_topology X ⊆ countable_complement_topology X.
Let U be given.
Assume HU: U ∈ finite_complement_topology X.
We prove the intermediate claim HUinPow: U ∈ 𝒫 X.
An exact proof term for the current goal is (SepE1 (𝒫 X) (λU0 : set ⇒ finite (X ∖ U0) ∨ U0 = Empty) U HU).
We prove the intermediate claim HUdata: finite (X ∖ U) ∨ U = Empty.
An exact proof term for the current goal is (SepE2 (𝒫 X) (λU0 : set ⇒ finite (X ∖ U0) ∨ U0 = Empty) U HU).
We prove the intermediate claim HUpred: countable (X ∖ U) ∨ U = Empty.
Apply HUdata (countable (X ∖ U) ∨ U = Empty) to the current goal.
Assume HUfin: finite (X ∖ U).
Apply orIL to the current goal.
An exact proof term for the current goal is (finite_countable (X ∖ U) HUfin).
Assume HUemp: U = Empty.
Apply orIR to the current goal.
An exact proof term for the current goal is HUemp.
An exact proof term for the current goal is (SepI (𝒫 X) (λU0 : set ⇒ countable (X ∖ U0) ∨ U0 = Empty) U HUinPow HUpred).
∎
Theorem. (finite_complement_coarser_than_discrete)
∀X : set, coarser_than (finite_complement_topology X) (discrete_topology X)
In Proofgold the corresponding term root is a66c14... and proposition id is 8e9d83...
Proof:
Let X be given.
We will prove finite_complement_topology X ⊆ discrete_topology X.
Let U be given.
Assume HU.
An exact proof term for the current goal is (SepE1 (𝒫 X) (λU0 : set ⇒ finite (X ∖ U0) ∨ U0 = Empty) U HU).
∎
Theorem. (indiscrete_coarser_than_countable_complement)
∀X : set, coarser_than (indiscrete_topology X) (countable_complement_topology X)
In Proofgold the corresponding term root is 8189cf... and proposition id is af77a9...
Proof:
Let X be given.
We will prove indiscrete_topology X ⊆ countable_complement_topology X.
Let U be given.
Assume HU: U ∈ indiscrete_topology X.
Apply UPairE U Empty X HU to the current goal.
Assume HUempty: U = Empty.
rewrite the current goal using HUempty (from left to right).
An exact proof term for the current goal is (countable_complement_topology_contains_empty X).
Assume HUX: U = X.
rewrite the current goal using HUX (from left to right).
An exact proof term for the current goal is (countable_complement_topology_contains_full X).
∎
Definition. We define finer_than_topology_by_inclusion to be λX T' T ⇒ topology_on X T' ∧ topology_on X T ∧ T ⊆ T' of type set → set → set → prop.
In Proofgold the corresponding term root is 0e75f6... and object id is b7d95b...
Theorem. (finer_than_topology_by_inclusion_eq)
∀X T' T : set, finer_than_topology X T' T ↔ finer_than_topology_by_inclusion X T' T
In Proofgold the corresponding term root is 0626b3... and proposition id is b0310e...
Proof:
Let X, T' and T be given.
Apply iffI to the current goal.
Assume H.
An exact proof term for the current goal is H.
Assume H.
An exact proof term for the current goal is H.
∎
Theorem. (lemma_union_of_topology_family_open)
∀X T UFam : set, topology_on X T → UFam ∈ 𝒫 T → ⋃ UFam ∈ T
In Proofgold the corresponding term root is 20b69b... and proposition id is 46a086...
Proof:
Let X, T and UFam be given.
Assume HT Hfam.
We prove the intermediate claim Hchunk1: ((T ⊆ 𝒫 X ∧ Empty ∈ T) ∧ X ∈ T) ∧ (∀UFam' ∈ 𝒫 T, ⋃ UFam' ∈ T).
An exact proof term for the current goal is (andEL (((T ⊆ 𝒫 X ∧ Empty ∈ T) ∧ X ∈ T) ∧ (∀UFam' ∈ 𝒫 T, ⋃ UFam' ∈ T)) (∀U ∈ T, ∀V ∈ T, U ∩ V ∈ T) HT).
We prove the intermediate claim Hunion_axiom: ∀UFam' ∈ 𝒫 T, ⋃ UFam' ∈ T.
An exact proof term for the current goal is (andER ((T ⊆ 𝒫 X ∧ Empty ∈ T) ∧ X ∈ T) (∀UFam' ∈ 𝒫 T, ⋃ UFam' ∈ T) Hchunk1).
An exact proof term for the current goal is (Hunion_axiom UFam Hfam).
∎
Theorem. (lemma_intersection_two_open)
∀X T U V : set, topology_on X T → U ∈ T → V ∈ T → U ∩ V ∈ T
In Proofgold the corresponding term root is 367ad1... and proposition id is 4e5488...
Proof:
Let X, T, U and V be given.
Assume HT HU HV.
We prove the intermediate claim Hax_inter: ∀U' ∈ T, ∀V' ∈ T, U' ∩ V' ∈ T.
An exact proof term for the current goal is (andER ((T ⊆ 𝒫 X ∧ Empty ∈ T ∧ X ∈ T ∧ (∀UFam ∈ 𝒫 T, ⋃ UFam ∈ T))) (∀U' ∈ T, ∀V' ∈ T, U' ∩ V' ∈ T) HT).
An exact proof term for the current goal is (Hax_inter U HU V HV).
∎
Definition. We define topological_space to be topology_on of type set → set → prop.
In Proofgold the corresponding term root is bb1861... and object id is 137515...
Definition. We define open_set_family to be λ_ T ⇒ T of type set → set → set.
In Proofgold the corresponding term root is 104bd1... and object id is 4b1310...
Definition. We define open_set to be λX T U ⇒ topology_on X T ∧ U ∈ T of type set → set → set → prop.
In Proofgold the corresponding term root is eedad1... and object id is 8d51ca...
Definition. We define basis_on to be λX B ⇒ B ⊆ 𝒫 X ∧ (∀x ∈ X, ∃b ∈ B, x ∈ b) ∧ (∀b1 ∈ B, ∀b2 ∈ B, ∀x : set, x ∈ b1 → x ∈ b2 → ∃b3 ∈ B, x ∈ b3 ∧ b3 ⊆ b1 ∩ b2) of type set → set → prop.
In Proofgold the corresponding term root is 4c6355... and object id is 2d1756...
Definition. We define generated_topology to be λX B ⇒ {U ∈ 𝒫 X|∀x ∈ U, ∃b ∈ B, x ∈ b ∧ b ⊆ U} of type set → set → set.
In Proofgold the corresponding term root is 4231ae... and object id is d0de89...
Axiom. (basis_elem_subset) We take the following as an axiom:
∀X B b : set, basis_on X B → b ∈ B → b ⊆ X
In Proofgold the corresponding term root is f6b3e3... and proposition id is 11dfc0...
Axiom. (basis_in_generated) We take the following as an axiom:
∀X B b : set, basis_on X B → b ∈ B → b ∈ generated_topology X B
In Proofgold the corresponding term root is fc5f07... and proposition id is 7d53ab...
Axiom. (generated_topology_subset) We take the following as an axiom:
∀X B U : set, U ∈ generated_topology X B → U ⊆ X
In Proofgold the corresponding term root is a7055b... and proposition id is 4e1fa4...
Theorem. (lemma_topology_from_basis)
∀X B : set, basis_on X B → topology_on X (generated_topology X B)
In Proofgold the corresponding term root is d7132e... and proposition id is 26d89d...
Proof:
Let X and B be given.
Assume HBasis.
We prove the intermediate claim HBleft: B ⊆ 𝒫 X ∧ (∀x ∈ X, ∃b ∈ B, x ∈ b).
An exact proof term for the current goal is (andEL (B ⊆ 𝒫 X ∧ (∀x ∈ X, ∃b ∈ B, x ∈ b)) (∀b1 ∈ B, ∀b2 ∈ B, ∀x : set, x ∈ b1 → x ∈ b2 → ∃b3 ∈ B, x ∈ b3 ∧ b3 ⊆ b1 ∩ b2) HBasis).
We prove the intermediate claim HBint: ∀b1 ∈ B, ∀b2 ∈ B, ∀x : set, x ∈ b1 → x ∈ b2 → ∃b3 ∈ B, x ∈ b3 ∧ b3 ⊆ b1 ∩ b2.
An exact proof term for the current goal is (andER (B ⊆ 𝒫 X ∧ (∀x ∈ X, ∃b ∈ B, x ∈ b)) (∀b1 ∈ B, ∀b2 ∈ B, ∀x : set, x ∈ b1 → x ∈ b2 → ∃b3 ∈ B, x ∈ b3 ∧ b3 ⊆ b1 ∩ b2) HBasis).
We prove the intermediate claim HBsub: B ⊆ 𝒫 X.
An exact proof term for the current goal is (andEL (B ⊆ 𝒫 X) (∀x ∈ X, ∃b ∈ B, x ∈ b) HBleft).
We prove the intermediate claim HBcov: ∀x ∈ X, ∃b ∈ B, x ∈ b.
An exact proof term for the current goal is (andER (B ⊆ 𝒫 X) (∀x ∈ X, ∃b ∈ B, x ∈ b) HBleft).
We prove the intermediate claim proofA: generated_topology X B ⊆ 𝒫 X.
Let U be given.
Assume HU: U ∈ generated_topology X B.
An exact proof term for the current goal is (SepE1 (𝒫 X) (λU0 : set ⇒ ∀x ∈ U0, ∃b ∈ B, x ∈ b ∧ b ⊆ U0) U HU).
We prove the intermediate claim proofB: Empty ∈ generated_topology X B.
An exact proof term for the current goal is (SepI (𝒫 X) (λU0 : set ⇒ ∀x ∈ U0, ∃b ∈ B, x ∈ b ∧ b ⊆ U0) Empty (Empty_In_Power X) (λx HxEmpty ⇒ EmptyE x HxEmpty (∃b ∈ B, x ∈ b ∧ b ⊆ Empty))).
We prove the intermediate claim proofC: X ∈ generated_topology X B.
We prove the intermediate claim HXprop: ∀x ∈ X, ∃b ∈ B, x ∈ b ∧ b ⊆ X.
Let x be given.
Assume HxX.
We prove the intermediate claim Hexb: ∃b ∈ B, x ∈ b.
An exact proof term for the current goal is (HBcov x HxX).
Apply Hexb to the current goal.
Let b be given.
Assume Hbpair.
We prove the intermediate claim HbB: b ∈ B.
An exact proof term for the current goal is (andEL (b ∈ B) (x ∈ b) Hbpair).
We prove the intermediate claim Hxb: x ∈ b.
An exact proof term for the current goal is (andER (b ∈ B) (x ∈ b) Hbpair).
We prove the intermediate claim HbsubX: b ⊆ X.
An exact proof term for the current goal is (PowerE X b (HBsub b HbB)).
We use b to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is HbB.
Apply andI to the current goal.
An exact proof term for the current goal is Hxb.
An exact proof term for the current goal is HbsubX.
An exact proof term for the current goal is (SepI (𝒫 X) (λU0 : set ⇒ ∀x ∈ U0, ∃b ∈ B, x ∈ b ∧ b ⊆ U0) X (Self_In_Power X) HXprop).
We prove the intermediate claim proofD: ∀UFam ∈ 𝒫 (generated_topology X B), ⋃ UFam ∈ generated_topology X B.
Let UFam be given.
Assume Hfam: UFam ∈ 𝒫 (generated_topology X B).
We prove the intermediate claim HsubFam: UFam ⊆ generated_topology X B.
An exact proof term for the current goal is (PowerE (generated_topology X B) UFam Hfam).
We prove the intermediate claim HPowUnion: ⋃ UFam ∈ 𝒫 X.
Apply PowerI X (⋃ UFam) to the current goal.
Let x be given.
Assume HxUnion.
Apply UnionE_impred UFam x HxUnion to the current goal.
Let U be given.
Assume HxU HUin.
We prove the intermediate claim HUtop: U ∈ generated_topology X B.
An exact proof term for the current goal is (HsubFam U HUin).
We prove the intermediate claim HUsubX: U ⊆ X.
An exact proof term for the current goal is (PowerE X U (SepE1 (𝒫 X) (λU0 : set ⇒ ∀x0 ∈ U0, ∃b ∈ B, x0 ∈ b ∧ b ⊆ U0) U HUtop)).
An exact proof term for the current goal is (HUsubX x HxU).
We prove the intermediate claim HUnionProp: ∀x ∈ ⋃ UFam, ∃b ∈ B, x ∈ b ∧ b ⊆ ⋃ UFam.
Let x be given.
Assume HxUnion.
Apply UnionE_impred UFam x HxUnion to the current goal.
Let U be given.
Assume HxU HUin.
We prove the intermediate claim HUtop: U ∈ generated_topology X B.
An exact proof term for the current goal is (HsubFam U HUin).
We prove the intermediate claim HUprop: ∀x0 ∈ U, ∃b ∈ B, x0 ∈ b ∧ b ⊆ U.
An exact proof term for the current goal is (SepE2 (𝒫 X) (λU0 : set ⇒ ∀x0 ∈ U0, ∃b ∈ B, x0 ∈ b ∧ b ⊆ U0) U HUtop).
We prove the intermediate claim Hexb: ∃b ∈ B, x ∈ b ∧ b ⊆ U.
An exact proof term for the current goal is (HUprop x HxU).
Apply Hexb to the current goal.
Let b be given.
Assume Hbpair.
We prove the intermediate claim HbB: b ∈ B.
An exact proof term for the current goal is (andEL (b ∈ B) (x ∈ b ∧ b ⊆ U) Hbpair).
We prove the intermediate claim Hbprop: x ∈ b ∧ b ⊆ U.
An exact proof term for the current goal is (andER (b ∈ B) (x ∈ b ∧ b ⊆ U) Hbpair).
We prove the intermediate claim Hxb: x ∈ b.
An exact proof term for the current goal is (andEL (x ∈ b) (b ⊆ U) Hbprop).
We prove the intermediate claim HbSubU: b ⊆ U.
An exact proof term for the current goal is (andER (x ∈ b) (b ⊆ U) Hbprop).
We use b to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is HbB.
Apply andI to the current goal.
An exact proof term for the current goal is Hxb.
Let y be given.
Assume Hyb.
Apply UnionI UFam y U (HbSubU y Hyb) HUin to the current goal.
An exact proof term for the current goal is (SepI (𝒫 X) (λU0 : set ⇒ ∀x ∈ U0, ∃b ∈ B, x ∈ b ∧ b ⊆ U0) (⋃ UFam) HPowUnion HUnionProp).
We prove the intermediate claim proofE: ∀U ∈ generated_topology X B, ∀V ∈ generated_topology X B, U ∩ V ∈ generated_topology X B.
Let U be given.
Assume HUtop.
Let V be given.
Assume HVtop.
We prove the intermediate claim HUprop: ∀x0 ∈ U, ∃b ∈ B, x0 ∈ b ∧ b ⊆ U.
An exact proof term for the current goal is (SepE2 (𝒫 X) (λU0 : set ⇒ ∀x0 ∈ U0, ∃b ∈ B, x0 ∈ b ∧ b ⊆ U0) U HUtop).
We prove the intermediate claim HVprop: ∀x0 ∈ V, ∃b ∈ B, x0 ∈ b ∧ b ⊆ V.
An exact proof term for the current goal is (SepE2 (𝒫 X) (λU0 : set ⇒ ∀x0 ∈ U0, ∃b ∈ B, x0 ∈ b ∧ b ⊆ U0) V HVtop).
We prove the intermediate claim HUsubX: U ⊆ X.
An exact proof term for the current goal is (PowerE X U (SepE1 (𝒫 X) (λU0 : set ⇒ ∀x0 ∈ U0, ∃b ∈ B, x0 ∈ b ∧ b ⊆ U0) U HUtop)).
We prove the intermediate claim HPowCap: U ∩ V ∈ 𝒫 X.
Apply PowerI X (U ∩ V) to the current goal.
Let x be given.
Assume HxCap.
Apply binintersectE U V x HxCap to the current goal.
Assume HxU HxV.
An exact proof term for the current goal is (HUsubX x HxU).
We prove the intermediate claim HCapProp: ∀x ∈ U ∩ V, ∃b ∈ B, x ∈ b ∧ b ⊆ U ∩ V.
Let x be given.
Assume HxCap.
Apply binintersectE U V x HxCap to the current goal.
Assume HxU HxV.
We prove the intermediate claim Hexb1: ∃b1 ∈ B, x ∈ b1 ∧ b1 ⊆ U.
An exact proof term for the current goal is (HUprop x HxU).
We prove the intermediate claim Hexb2: ∃b2 ∈ B, x ∈ b2 ∧ b2 ⊆ V.
An exact proof term for the current goal is (HVprop x HxV).
Apply Hexb1 to the current goal.
Let b1 be given.
Assume Hbpair1.
We prove the intermediate claim Hb1: b1 ∈ B.
An exact proof term for the current goal is (andEL (b1 ∈ B) (x ∈ b1 ∧ b1 ⊆ U) Hbpair1).
We prove the intermediate claim Hb1prop: x ∈ b1 ∧ b1 ⊆ U.
An exact proof term for the current goal is (andER (b1 ∈ B) (x ∈ b1 ∧ b1 ⊆ U) Hbpair1).
We prove the intermediate claim Hb1x: x ∈ b1.
An exact proof term for the current goal is (andEL (x ∈ b1) (b1 ⊆ U) Hb1prop).
We prove the intermediate claim Hb1Sub: b1 ⊆ U.
An exact proof term for the current goal is (andER (x ∈ b1) (b1 ⊆ U) Hb1prop).
Apply Hexb2 to the current goal.
Let b2 be given.
Assume Hbpair2.
We prove the intermediate claim Hb2: b2 ∈ B.
An exact proof term for the current goal is (andEL (b2 ∈ B) (x ∈ b2 ∧ b2 ⊆ V) Hbpair2).
We prove the intermediate claim Hb2prop: x ∈ b2 ∧ b2 ⊆ V.
An exact proof term for the current goal is (andER (b2 ∈ B) (x ∈ b2 ∧ b2 ⊆ V) Hbpair2).
We prove the intermediate claim Hb2x: x ∈ b2.
An exact proof term for the current goal is (andEL (x ∈ b2) (b2 ⊆ V) Hb2prop).
We prove the intermediate claim Hb2Sub: b2 ⊆ V.
An exact proof term for the current goal is (andER (x ∈ b2) (b2 ⊆ V) Hb2prop).
We prove the intermediate claim Hexb3: ∃b3 ∈ B, x ∈ b3 ∧ b3 ⊆ b1 ∩ b2.
An exact proof term for the current goal is (HBint b1 Hb1 b2 Hb2 x Hb1x Hb2x).
Apply Hexb3 to the current goal.
Let b3 be given.
Assume Hbpair3.
We prove the intermediate claim Hb3: b3 ∈ B.
An exact proof term for the current goal is (andEL (b3 ∈ B) (x ∈ b3 ∧ b3 ⊆ b1 ∩ b2) Hbpair3).
We prove the intermediate claim Hb3prop: x ∈ b3 ∧ b3 ⊆ b1 ∩ b2.
An exact proof term for the current goal is (andER (b3 ∈ B) (x ∈ b3 ∧ b3 ⊆ b1 ∩ b2) Hbpair3).
We prove the intermediate claim HxB3: x ∈ b3.
An exact proof term for the current goal is (andEL (x ∈ b3) (b3 ⊆ b1 ∩ b2) Hb3prop).
We prove the intermediate claim Hb3Sub: b3 ⊆ b1 ∩ b2.
An exact proof term for the current goal is (andER (x ∈ b3) (b3 ⊆ b1 ∩ b2) Hb3prop).
We use b3 to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is Hb3.
Apply andI to the current goal.
An exact proof term for the current goal is HxB3.
Let y be given.
Assume Hyb3.
We prove the intermediate claim Hy_in_b1b2: y ∈ b1 ∩ b2.
An exact proof term for the current goal is (Hb3Sub y Hyb3).
Apply binintersectE b1 b2 y Hy_in_b1b2 to the current goal.
Assume Hyb1 Hyb2.
Apply binintersectI U V y (Hb1Sub y Hyb1) (Hb2Sub y Hyb2) to the current goal.
An exact proof term for the current goal is (SepI (𝒫 X) (λU0 : set ⇒ ∀x0 ∈ U0, ∃b ∈ B, x0 ∈ b ∧ b ⊆ U0) (U ∩ V) HPowCap HCapProp).
We will prove generated_topology X B ⊆ 𝒫 X ∧ Empty ∈ generated_topology X B ∧ X ∈ generated_topology X B ∧ (∀UFam ∈ 𝒫 (generated_topology X B), ⋃ UFam ∈ generated_topology X B) ∧ (∀U ∈ generated_topology X B, ∀V ∈ generated_topology X B, U ∩ V ∈ generated_topology X B).
Apply andI to the current goal.
Apply andI to the current goal.
Apply andI to the current goal.
Apply andI to the current goal.
An exact proof term for the current goal is proofA.
An exact proof term for the current goal is proofB.
An exact proof term for the current goal is proofC.
An exact proof term for the current goal is proofD.
An exact proof term for the current goal is proofE.
∎
Theorem. (generated_topology_contains_basis)
∀X B : set, basis_on X B → ∀b : set, b ∈ B → b ∈ generated_topology X B
In Proofgold the corresponding term root is 644e5a... and proposition id is 4df7b9...
Proof:
Let X and B be given.
Assume HBasis.
We prove the intermediate claim HBsub: B ⊆ 𝒫 X.
An exact proof term for the current goal is (andEL (B ⊆ 𝒫 X) (∀x ∈ X, ∃b ∈ B, x ∈ b) (andEL (B ⊆ 𝒫 X ∧ (∀x ∈ X, ∃b ∈ B, x ∈ b)) (∀b1 ∈ B, ∀b2 ∈ B, ∀x : set, x ∈ b1 → x ∈ b2 → ∃b3 ∈ B, x ∈ b3 ∧ b3 ⊆ b1 ∩ b2) HBasis)).
We prove the intermediate claim HBint: ∀b1 ∈ B, ∀b2 ∈ B, ∀x : set, x ∈ b1 → x ∈ b2 → ∃b3 ∈ B, x ∈ b3 ∧ b3 ⊆ b1 ∩ b2.
An exact proof term for the current goal is (andER (B ⊆ 𝒫 X ∧ (∀x ∈ X, ∃b ∈ B, x ∈ b)) (∀b1 ∈ B, ∀b2 ∈ B, ∀x : set, x ∈ b1 → x ∈ b2 → ∃b3 ∈ B, x ∈ b3 ∧ b3 ⊆ b1 ∩ b2) HBasis).
Let b0 be given.
Assume Hb0.
We prove the intermediate claim Hb0_subX: b0 ⊆ X.
An exact proof term for the current goal is (PowerE X b0 (HBsub b0 Hb0)).
We prove the intermediate claim Hb0prop: ∀x ∈ b0, ∃b ∈ B, x ∈ b ∧ b ⊆ b0.
Let x be given.
Assume Hxb0.
We prove the intermediate claim Hexb3: ∃b3 ∈ B, x ∈ b3 ∧ b3 ⊆ b0 ∩ b0.
An exact proof term for the current goal is (HBint b0 Hb0 b0 Hb0 x Hxb0 Hxb0).
Apply Hexb3 to the current goal.
Let b3 be given.
Assume Hb3pair.
We prove the intermediate claim Hb3: b3 ∈ B.
An exact proof term for the current goal is (andEL (b3 ∈ B) (x ∈ b3 ∧ b3 ⊆ b0 ∩ b0) Hb3pair).
We prove the intermediate claim Hb3prop: x ∈ b3 ∧ b3 ⊆ b0 ∩ b0.
An exact proof term for the current goal is (andER (b3 ∈ B) (x ∈ b3 ∧ b3 ⊆ b0 ∩ b0) Hb3pair).
We prove the intermediate claim Hxb3: x ∈ b3.
An exact proof term for the current goal is (andEL (x ∈ b3) (b3 ⊆ b0 ∩ b0) Hb3prop).
We prove the intermediate claim Hb3sub_inter: b3 ⊆ b0 ∩ b0.
An exact proof term for the current goal is (andER (x ∈ b3) (b3 ⊆ b0 ∩ b0) Hb3prop).
We prove the intermediate claim Hb3subb0: b3 ⊆ b0.
Let y be given.
Assume Hyb3.
We prove the intermediate claim Hycap: y ∈ b0 ∩ b0.
An exact proof term for the current goal is (Hb3sub_inter y Hyb3).
Apply binintersectE b0 b0 y Hycap to the current goal.
Assume Hy1 Hy2.
An exact proof term for the current goal is Hy1.
We use b3 to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is Hb3.
Apply andI to the current goal.
An exact proof term for the current goal is Hxb3.
An exact proof term for the current goal is Hb3subb0.
An exact proof term for the current goal is (SepI (𝒫 X) (λU0 : set ⇒ ∀x ∈ U0, ∃b ∈ B, x ∈ b ∧ b ⊆ U0) b0 (PowerI X b0 Hb0_subX) Hb0prop).
∎
Definition. We define basis_generates to be λX B T ⇒ basis_on X B ∧ generated_topology X B = T of type set → set → set → prop.
In Proofgold the corresponding term root is 6cce2e... and object id is 8c2cd7...
Definition. We define basis_refines to be λX B T ⇒ topology_on X T ∧ (∀U ∈ T, ∀x ∈ U, ∃b ∈ B, x ∈ b ∧ b ⊆ U) of type set → set → set → prop.
In Proofgold the corresponding term root is 2cf9a4... and object id is e74588...
Theorem. (lemma_generated_topology_characterization)
∀X B : set, basis_on X B → generated_topology X B = {U ∈ 𝒫 X|∀x ∈ U, ∃b ∈ B, x ∈ b ∧ b ⊆ U}
In Proofgold the corresponding term root is c8432c... and proposition id is 063588...
Proof:
Let X and B be given.
Assume HBasis.
Use reflexivity.
∎
Theorem. (open_sets_as_unions_of_basis)
∀X B : set, basis_on X B → ∀U : set, open_in X (generated_topology X B) U → ∃Fam ∈ 𝒫 B, ⋃ Fam = U
In Proofgold the corresponding term root is c1e863... and proposition id is d2093b...
Proof:
Let X and B be given.
Assume HBasis.
We prove the intermediate claim HBsub: B ⊆ 𝒫 X.
An exact proof term for the current goal is (andEL (B ⊆ 𝒫 X) (∀x ∈ X, ∃b ∈ B, x ∈ b) (andEL (B ⊆ 𝒫 X ∧ (∀x ∈ X, ∃b ∈ B, x ∈ b)) (∀b1 ∈ B, ∀b2 ∈ B, ∀x : set, x ∈ b1 → x ∈ b2 → ∃b3 ∈ B, x ∈ b3 ∧ b3 ⊆ b1 ∩ b2) HBasis)).
Let U be given.
Assume HUopen.
We prove the intermediate claim HUtop: U ∈ generated_topology X B.
An exact proof term for the current goal is (andER (topology_on X (generated_topology X B)) (U ∈ generated_topology X B) HUopen).
We prove the intermediate claim HUprop: ∀x ∈ U, ∃b ∈ B, x ∈ b ∧ b ⊆ U.
An exact proof term for the current goal is (SepE2 (𝒫 X) (λU0 : set ⇒ ∀x0 ∈ U0, ∃b ∈ B, x0 ∈ b ∧ b ⊆ U0) U HUtop).
Set Fam to be the term {b ∈ B|b ⊆ U} of type set.
We prove the intermediate claim HFamPow: Fam ∈ 𝒫 B.
Apply PowerI B Fam to the current goal.
Let b be given.
Assume HbFam.
An exact proof term for the current goal is (SepE1 B (λb0 : set ⇒ b0 ⊆ U) b HbFam).
We prove the intermediate claim HUnion_eq: ⋃ Fam = U.
Apply set_ext to the current goal.
Let x be given.
Assume HxUnion.
Apply UnionE_impred Fam x HxUnion to the current goal.
Let b be given.
Assume Hxb HbFam.
We prove the intermediate claim HbsubU: b ⊆ U.
An exact proof term for the current goal is (SepE2 B (λb0 : set ⇒ b0 ⊆ U) b HbFam).
An exact proof term for the current goal is (HbsubU x Hxb).
Let x be given.
Assume HxU.
We prove the intermediate claim Hexb: ∃b ∈ B, x ∈ b ∧ b ⊆ U.
An exact proof term for the current goal is (HUprop x HxU).
Apply Hexb to the current goal.
Let b be given.
Assume Hbpair.
We prove the intermediate claim HbB: b ∈ B.
An exact proof term for the current goal is (andEL (b ∈ B) (x ∈ b ∧ b ⊆ U) Hbpair).
We prove the intermediate claim Hbprop: x ∈ b ∧ b ⊆ U.
An exact proof term for the current goal is (andER (b ∈ B) (x ∈ b ∧ b ⊆ U) Hbpair).
We prove the intermediate claim Hxb: x ∈ b.
An exact proof term for the current goal is (andEL (x ∈ b) (b ⊆ U) Hbprop).
We prove the intermediate claim HbsubU: b ⊆ U.
An exact proof term for the current goal is (andER (x ∈ b) (b ⊆ U) Hbprop).
We prove the intermediate claim HbFam: b ∈ Fam.
An exact proof term for the current goal is (SepI B (λb0 : set ⇒ b0 ⊆ U) b HbB HbsubU).
An exact proof term for the current goal is (UnionI Fam x b Hxb HbFam).
We use Fam to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is HFamPow.
An exact proof term for the current goal is HUnion_eq.
∎
Theorem. (basis_generates_open_sets)
∀X B : set, basis_on X B → ∀U : set, (∃Fam ∈ 𝒫 B, ⋃ Fam = U) → open_in X (generated_topology X B) U
In Proofgold the corresponding term root is adf14f... and proposition id is ab7b94...
Proof:
Let X and B be given.
Assume HBasis.
We prove the intermediate claim HBsub: B ⊆ 𝒫 X.
An exact proof term for the current goal is (andEL (B ⊆ 𝒫 X) (∀x ∈ X, ∃b ∈ B, x ∈ b) (andEL (B ⊆ 𝒫 X ∧ (∀x ∈ X, ∃b ∈ B, x ∈ b)) (∀b1 ∈ B, ∀b2 ∈ B, ∀x : set, x ∈ b1 → x ∈ b2 → ∃b3 ∈ B, x ∈ b3 ∧ b3 ⊆ b1 ∩ b2) HBasis)).
Let U be given.
Assume Hex.
We prove the intermediate claim HUGen: U ∈ generated_topology X B.
Apply Hex to the current goal.
Let Fam be given.
Assume HFampair.
We prove the intermediate claim HFamPow: Fam ∈ 𝒫 B.
An exact proof term for the current goal is (andEL (Fam ∈ 𝒫 B) (⋃ Fam = U) HFampair).
We prove the intermediate claim HUnionEq: ⋃ Fam = U.
An exact proof term for the current goal is (andER (Fam ∈ 𝒫 B) (⋃ Fam = U) HFampair).
We prove the intermediate claim HFamSubB: Fam ⊆ B.
An exact proof term for the current goal is (PowerE B Fam HFamPow).
We prove the intermediate claim HFamSubX: Fam ⊆ 𝒫 X.
Let b be given.
Assume HbFam.
We prove the intermediate claim HbB: b ∈ B.
An exact proof term for the current goal is (HFamSubB b HbFam).
An exact proof term for the current goal is (HBsub b HbB).
We prove the intermediate claim HUnionSubX: ⋃ Fam ⊆ X.
Let x be given.
Assume HxUnion.
Apply UnionE_impred Fam x HxUnion to the current goal.
Let b be given.
Assume Hxb HbFam.
We prove the intermediate claim HbSubX: b ⊆ X.
An exact proof term for the current goal is (PowerE X b (HFamSubX b HbFam)).
An exact proof term for the current goal is (HbSubX x Hxb).
We prove the intermediate claim HUnionSubU: ⋃ Fam ⊆ U.
rewrite the current goal using HUnionEq (from left to right).
An exact proof term for the current goal is (Subq_ref U).
We prove the intermediate claim HUsubUnion: U ⊆ ⋃ Fam.
rewrite the current goal using HUnionEq (from right to left).
An exact proof term for the current goal is (Subq_ref (⋃ Fam)).
We prove the intermediate claim HUsubX: U ⊆ X.
An exact proof term for the current goal is (Subq_tra U (⋃ Fam) X HUsubUnion HUnionSubX).
We prove the intermediate claim HUpropU: ∀x ∈ U, ∃b ∈ B, x ∈ b ∧ b ⊆ U.
Let x be given.
Assume HxU.
We prove the intermediate claim HxUnion: x ∈ ⋃ Fam.
An exact proof term for the current goal is (HUsubUnion x HxU).
Apply UnionE_impred Fam x HxUnion to the current goal.
Let b be given.
Assume Hxb HbFam.
We prove the intermediate claim HbB: b ∈ B.
An exact proof term for the current goal is (HFamSubB b HbFam).
We prove the intermediate claim HbSubUnion: b ⊆ ⋃ Fam.
Let y be given.
Assume Hyb.
An exact proof term for the current goal is (UnionI Fam y b Hyb HbFam).
We prove the intermediate claim HbSubU: b ⊆ U.
An exact proof term for the current goal is (Subq_tra b (⋃ Fam) U HbSubUnion HUnionSubU).
We use b to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is HbB.
Apply andI to the current goal.
An exact proof term for the current goal is Hxb.
An exact proof term for the current goal is HbSubU.
An exact proof term for the current goal is (SepI (𝒫 X) (λU0 : set ⇒ ∀x0 ∈ U0, ∃b0 ∈ B, x0 ∈ b0 ∧ b0 ⊆ U0) U (PowerI X U HUsubX) HUpropU).
An exact proof term for the current goal is (andI (topology_on X (generated_topology X B)) (U ∈ generated_topology X B) (lemma_topology_from_basis X B HBasis) HUGen).
∎
Theorem. (open_as_union_of_basis_elements)
∀X B : set, basis_on X B → ∀U : set, open_in X (generated_topology X B) U → U = ⋃ {b ∈ B|b ⊆ U}
In Proofgold the corresponding term root is 3b6dba... and proposition id is 0c265f...
Proof:
Let X and B be given.
Assume HBasis.
Let U be given.
Assume HUopen.
We prove the intermediate claim HUtop: U ∈ generated_topology X B.
An exact proof term for the current goal is (andER (topology_on X (generated_topology X B)) (U ∈ generated_topology X B) HUopen).
We prove the intermediate claim HUprop: ∀x ∈ U, ∃b ∈ B, x ∈ b ∧ b ⊆ U.
An exact proof term for the current goal is (SepE2 (𝒫 X) (λU0 : set ⇒ ∀x0 ∈ U0, ∃b ∈ B, x0 ∈ b ∧ b ⊆ U0) U HUtop).
Set Fam to be the term {b ∈ B|b ⊆ U} of type set.
Apply set_ext to the current goal.
Let x be given.
Assume HxU.
We prove the intermediate claim Hexb: ∃b ∈ B, x ∈ b ∧ b ⊆ U.
An exact proof term for the current goal is (HUprop x HxU).
Apply Hexb to the current goal.
Let b be given.
Assume Hbpair.
We prove the intermediate claim HbB: b ∈ B.
An exact proof term for the current goal is (andEL (b ∈ B) (x ∈ b ∧ b ⊆ U) Hbpair).
We prove the intermediate claim Hbprop: x ∈ b ∧ b ⊆ U.
An exact proof term for the current goal is (andER (b ∈ B) (x ∈ b ∧ b ⊆ U) Hbpair).
We prove the intermediate claim Hxb: x ∈ b.
An exact proof term for the current goal is (andEL (x ∈ b) (b ⊆ U) Hbprop).
We prove the intermediate claim HbsubU: b ⊆ U.
An exact proof term for the current goal is (andER (x ∈ b) (b ⊆ U) Hbprop).
We prove the intermediate claim HbFam: b ∈ Fam.
An exact proof term for the current goal is (SepI B (λb0 : set ⇒ b0 ⊆ U) b HbB HbsubU).
An exact proof term for the current goal is (UnionI Fam x b Hxb HbFam).
Let x be given.
Assume HxUnion.
Apply UnionE_impred Fam x HxUnion to the current goal.
Let b be given.
Assume Hxb HbFam.
We prove the intermediate claim HbsubU: b ⊆ U.
An exact proof term for the current goal is (SepE2 B (λb0 : set ⇒ b0 ⊆ U) b HbFam).
An exact proof term for the current goal is (HbsubU x Hxb).
∎
Theorem. (basis_refines_topology)
∀X T C : set, topology_on X T → (∀c ∈ C, c ∈ T) → (∀U ∈ T, ∀x ∈ U, ∃Cx ∈ C, x ∈ Cx ∧ Cx ⊆ U) → basis_on X C ∧ generated_topology X C = T
In Proofgold the corresponding term root is 8a26a8... and proposition id is ced1f4...
Proof:
Let X, T and C be given.
Assume Htop HCsub Href.
We prove the intermediate claim Hleft: ((T ⊆ 𝒫 X ∧ Empty ∈ T) ∧ X ∈ T) ∧ (∀UFam ∈ 𝒫 T, ⋃ UFam ∈ T).
An exact proof term for the current goal is (andEL (((T ⊆ 𝒫 X ∧ Empty ∈ T) ∧ X ∈ T) ∧ (∀UFam ∈ 𝒫 T, ⋃ UFam ∈ T)) (∀U ∈ T, ∀V ∈ T, U ∩ V ∈ T) Htop).
We prove the intermediate claim Hcore: (T ⊆ 𝒫 X ∧ Empty ∈ T) ∧ X ∈ T.
An exact proof term for the current goal is (andEL ((T ⊆ 𝒫 X ∧ Empty ∈ T) ∧ X ∈ T) (∀UFam ∈ 𝒫 T, ⋃ UFam ∈ T) Hleft).
We prove the intermediate claim HTPowEmpty: T ⊆ 𝒫 X ∧ Empty ∈ T.
An exact proof term for the current goal is (andEL (T ⊆ 𝒫 X ∧ Empty ∈ T) (X ∈ T) Hcore).
We prove the intermediate claim HTsubPow: T ⊆ 𝒫 X.
An exact proof term for the current goal is (andEL (T ⊆ 𝒫 X) (Empty ∈ T) HTPowEmpty).
We prove the intermediate claim HXT: X ∈ T.
An exact proof term for the current goal is (andER (T ⊆ 𝒫 X ∧ Empty ∈ T) (X ∈ T) Hcore).
We prove the intermediate claim HUnionClosed: ∀UFam ∈ 𝒫 T, ⋃ UFam ∈ T.
An exact proof term for the current goal is (andER ((T ⊆ 𝒫 X ∧ Empty ∈ T) ∧ X ∈ T) (∀UFam ∈ 𝒫 T, ⋃ UFam ∈ T) Hleft).
We prove the intermediate claim HInterClosed: ∀U ∈ T, ∀V ∈ T, U ∩ V ∈ T.
An exact proof term for the current goal is (andER (((T ⊆ 𝒫 X ∧ Empty ∈ T) ∧ X ∈ T) ∧ (∀UFam ∈ 𝒫 T, ⋃ UFam ∈ T)) (∀U ∈ T, ∀V ∈ T, U ∩ V ∈ T) Htop).
We prove the intermediate claim HBasis: basis_on X C.
We will prove (C ⊆ 𝒫 X ∧ (∀x ∈ X, ∃c ∈ C, x ∈ c) ∧ (∀b1 ∈ C, ∀b2 ∈ C, ∀x : set, x ∈ b1 → x ∈ b2 → ∃b3 ∈ C, x ∈ b3 ∧ b3 ⊆ b1 ∩ b2)).
Apply andI to the current goal.
Apply andI to the current goal.
Let c be given.
Assume HcC.
An exact proof term for the current goal is (HTsubPow c (HCsub c HcC)).
Let x be given.
Assume HxX.
We prove the intermediate claim Hex: ∃c ∈ C, x ∈ c ∧ c ⊆ X.
An exact proof term for the current goal is (Href X HXT x HxX).
Apply Hex to the current goal.
Let c be given.
Assume Hpair.
We prove the intermediate claim HcC: c ∈ C.
An exact proof term for the current goal is (andEL (c ∈ C) (x ∈ c ∧ c ⊆ X) Hpair).
We prove the intermediate claim Hcprop: x ∈ c ∧ c ⊆ X.
An exact proof term for the current goal is (andER (c ∈ C) (x ∈ c ∧ c ⊆ X) Hpair).
We prove the intermediate claim Hxc: x ∈ c.
An exact proof term for the current goal is (andEL (x ∈ c) (c ⊆ X) Hcprop).
We use c to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is HcC.
An exact proof term for the current goal is Hxc.
Let c1 be given.
Assume Hc1C.
Let c2 be given.
Assume Hc2C.
Let x be given.
Assume Hxc1 Hxc2.
We prove the intermediate claim Hc1T: c1 ∈ T.
An exact proof term for the current goal is (HCsub c1 Hc1C).
We prove the intermediate claim Hc2T: c2 ∈ T.
An exact proof term for the current goal is (HCsub c2 Hc2C).
We prove the intermediate claim HcapT: c1 ∩ c2 ∈ T.
An exact proof term for the current goal is (HInterClosed c1 Hc1T c2 Hc2T).
We prove the intermediate claim HxCap: x ∈ c1 ∩ c2.
An exact proof term for the current goal is (binintersectI c1 c2 x Hxc1 Hxc2).
We prove the intermediate claim Hex: ∃c3 ∈ C, x ∈ c3 ∧ c3 ⊆ c1 ∩ c2.
An exact proof term for the current goal is (Href (c1 ∩ c2) HcapT x HxCap).
An exact proof term for the current goal is Hex.
We prove the intermediate claim Hgen_sub_T: generated_topology X C ⊆ T.
Let U be given.
Assume HUgen: U ∈ generated_topology X C.
We prove the intermediate claim HUsubX: U ⊆ X.
An exact proof term for the current goal is (PowerE X U (SepE1 (𝒫 X) (λU0 : set ⇒ ∀x0 ∈ U0, ∃b0 ∈ C, x0 ∈ b0 ∧ b0 ⊆ U0) U HUgen)).
We prove the intermediate claim HUprop: ∀x ∈ U, ∃c ∈ C, x ∈ c ∧ c ⊆ U.
An exact proof term for the current goal is (SepE2 (𝒫 X) (λU0 : set ⇒ ∀x0 ∈ U0, ∃b0 ∈ C, x0 ∈ b0 ∧ b0 ⊆ U0) U HUgen).
Set Fam to be the term {c ∈ C|c ⊆ U} of type set.
We prove the intermediate claim HFamSubC: Fam ⊆ C.
Let c be given.
Assume HcFam.
An exact proof term for the current goal is (SepE1 C (λc0 : set ⇒ c0 ⊆ U) c HcFam).
We prove the intermediate claim HFamSubT: Fam ⊆ T.
Let c be given.
Assume HcFam.
We prove the intermediate claim HcC: c ∈ C.
An exact proof term for the current goal is (HFamSubC c HcFam).
An exact proof term for the current goal is (HCsub c HcC).
We prove the intermediate claim HFamPowT: Fam ∈ 𝒫 T.
An exact proof term for the current goal is (PowerI T Fam HFamSubT).
We prove the intermediate claim HUnionSubU: ⋃ Fam ⊆ U.
Let x be given.
Assume HxUnion.
Apply UnionE_impred Fam x HxUnion to the current goal.
Let c be given.
Assume Hxc HcFam.
We prove the intermediate claim Hcprop: c ⊆ U.
An exact proof term for the current goal is (SepE2 C (λc0 : set ⇒ c0 ⊆ U) c HcFam).
An exact proof term for the current goal is (Hcprop x Hxc).
We prove the intermediate claim HUsubUnion: U ⊆ ⋃ Fam.
Let x be given.
Assume HxU.
We prove the intermediate claim Hex: ∃c ∈ C, x ∈ c ∧ c ⊆ U.
An exact proof term for the current goal is (HUprop x HxU).
Apply Hex to the current goal.
Let c be given.
Assume Hcpair.
We prove the intermediate claim HcC: c ∈ C.
An exact proof term for the current goal is (andEL (c ∈ C) (x ∈ c ∧ c ⊆ U) Hcpair).
We prove the intermediate claim Hcprop: x ∈ c ∧ c ⊆ U.
An exact proof term for the current goal is (andER (c ∈ C) (x ∈ c ∧ c ⊆ U) Hcpair).
We prove the intermediate claim Hxc: x ∈ c.
An exact proof term for the current goal is (andEL (x ∈ c) (c ⊆ U) Hcprop).
We prove the intermediate claim HcsubU: c ⊆ U.
An exact proof term for the current goal is (andER (x ∈ c) (c ⊆ U) Hcprop).
We prove the intermediate claim HcFam: c ∈ Fam.
An exact proof term for the current goal is (SepI C (λc0 : set ⇒ c0 ⊆ U) c HcC HcsubU).
An exact proof term for the current goal is (UnionI Fam x c Hxc HcFam).
We prove the intermediate claim HUnionEqU: ⋃ Fam = U.
Apply set_ext to the current goal.
Let x be given.
Assume HxUnion.
An exact proof term for the current goal is (HUnionSubU x HxUnion).
Let x be given.
Assume HxU.
An exact proof term for the current goal is (HUsubUnion x HxU).
We prove the intermediate claim HUnionInT: ⋃ Fam ∈ T.
An exact proof term for the current goal is (HUnionClosed Fam HFamPowT).
rewrite the current goal using HUnionEqU (from right to left).
An exact proof term for the current goal is HUnionInT.
We prove the intermediate claim HT_sub_gen: T ⊆ generated_topology X C.
Let U be given.
Assume HU: U ∈ T.
We prove the intermediate claim HUsubX: U ⊆ X.
An exact proof term for the current goal is (PowerE X U (HTsubPow U HU)).
We prove the intermediate claim HUprop: ∀x ∈ U, ∃c ∈ C, x ∈ c ∧ c ⊆ U.
Let x be given.
Assume HxU.
An exact proof term for the current goal is (Href U HU x HxU).
An exact proof term for the current goal is (SepI (𝒫 X) (λU0 : set ⇒ ∀x0 ∈ U0, ∃b0 ∈ C, x0 ∈ b0 ∧ b0 ⊆ U0) U (PowerI X U HUsubX) HUprop).
We prove the intermediate claim HEqual: generated_topology X C = T.
Apply set_ext to the current goal.
Let U be given.
Assume HUgen.
An exact proof term for the current goal is (Hgen_sub_T U HUgen).
Let U be given.
Assume HU.
An exact proof term for the current goal is (HT_sub_gen U HU).
Apply andI to the current goal.
An exact proof term for the current goal is HBasis.
An exact proof term for the current goal is HEqual.
∎
Theorem. (lemma13_2_basis_from_open_subcollection)
∀X T C : set, topology_on X T → (∀c ∈ C, c ∈ T) → (∀U ∈ T, ∀x ∈ U, ∃c ∈ C, x ∈ c ∧ c ⊆ U) → basis_on X C ∧ generated_topology X C = T
In Proofgold the corresponding term root is 8a26a8... and proposition id is ced1f4...
Proof:
Let X, T and C be given.
Assume Htop HCsub Href.
An exact proof term for the current goal is (basis_refines_topology X T C Htop HCsub Href).
∎
Theorem. (finer_via_basis)
∀X B B' : set, basis_on X B → basis_on X B' → (∀x ∈ X, ∀b : set, b ∈ B → x ∈ b → ∃b' ∈ B', x ∈ b' ∧ b' ⊆ b) → finer_than (generated_topology X B') (generated_topology X B)
In Proofgold the corresponding term root is f5076c... and proposition id is 339bed...
Proof:
Let X, B and B' be given.
Assume HB HB' Hcond.
We prove the intermediate claim HT: topology_on X (generated_topology X B).
An exact proof term for the current goal is (lemma_topology_from_basis X B HB).
We prove the intermediate claim HRefProp: ∀U ∈ generated_topology X B, ∀x ∈ U, ∃b' ∈ B', x ∈ b' ∧ b' ⊆ U.
Let U be given.
Assume HU: U ∈ generated_topology X B.
Let x be given.
Assume HxU.
We prove the intermediate claim HUsubX: U ⊆ X.
An exact proof term for the current goal is (PowerE X U (SepE1 (𝒫 X) (λU0 : set ⇒ ∀x0 ∈ U0, ∃b ∈ B, x0 ∈ b ∧ b ⊆ U0) U HU)).
We prove the intermediate claim HxX: x ∈ X.
An exact proof term for the current goal is (HUsubX x HxU).
We prove the intermediate claim HUprop: ∀x0 ∈ U, ∃b ∈ B, x0 ∈ b ∧ b ⊆ U.
An exact proof term for the current goal is (SepE2 (𝒫 X) (λU0 : set ⇒ ∀x0 ∈ U0, ∃b ∈ B, x0 ∈ b ∧ b ⊆ U0) U HU).
We prove the intermediate claim Hexb: ∃b ∈ B, x ∈ b ∧ b ⊆ U.
An exact proof term for the current goal is (HUprop x HxU).
Apply Hexb to the current goal.
Let b be given.
Assume Hbpair.
We prove the intermediate claim HbB: b ∈ B.
An exact proof term for the current goal is (andEL (b ∈ B) (x ∈ b ∧ b ⊆ U) Hbpair).
We prove the intermediate claim Hbprop: x ∈ b ∧ b ⊆ U.
An exact proof term for the current goal is (andER (b ∈ B) (x ∈ b ∧ b ⊆ U) Hbpair).
We prove the intermediate claim Hxb: x ∈ b.
An exact proof term for the current goal is (andEL (x ∈ b) (b ⊆ U) Hbprop).
We prove the intermediate claim HbsubU: b ⊆ U.
An exact proof term for the current goal is (andER (x ∈ b) (b ⊆ U) Hbprop).
We prove the intermediate claim Hexb': ∃b' ∈ B', x ∈ b' ∧ b' ⊆ b.
An exact proof term for the current goal is (Hcond x HxX b HbB Hxb).
Apply Hexb' to the current goal.
Let b' be given.
Assume Hb'pair.
We prove the intermediate claim Hb'B: b' ∈ B'.
An exact proof term for the current goal is (andEL (b' ∈ B') (x ∈ b' ∧ b' ⊆ b) Hb'pair).
We prove the intermediate claim Hb'prop: x ∈ b' ∧ b' ⊆ b.
An exact proof term for the current goal is (andER (b' ∈ B') (x ∈ b' ∧ b' ⊆ b) Hb'pair).
We prove the intermediate claim Hxb': x ∈ b'.
An exact proof term for the current goal is (andEL (x ∈ b') (b' ⊆ b) Hb'prop).
We prove the intermediate claim Hb'subb: b' ⊆ b.
An exact proof term for the current goal is (andER (x ∈ b') (b' ⊆ b) Hb'prop).
We use b' to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is Hb'B.
Apply andI to the current goal.
An exact proof term for the current goal is Hxb'.
An exact proof term for the current goal is (Subq_tra b' b U Hb'subb HbsubU).
We will prove generated_topology X B ⊆ generated_topology X B'.
Let U be given.
Assume HU.
We prove the intermediate claim HUsubX: U ⊆ X.
An exact proof term for the current goal is (PowerE X U (SepE1 (𝒫 X) (λU0 : set ⇒ ∀x0 ∈ U0, ∃b0 ∈ B, x0 ∈ b0 ∧ b0 ⊆ U0) U HU)).
We prove the intermediate claim HUprop: ∀x ∈ U, ∃b' ∈ B', x ∈ b' ∧ b' ⊆ U.
An exact proof term for the current goal is (HRefProp U HU).
An exact proof term for the current goal is (SepI (𝒫 X) (λU0 : set ⇒ ∀x0 ∈ U0, ∃b0 ∈ B', x0 ∈ b0 ∧ b0 ⊆ U0) U (PowerI X U HUsubX) HUprop).
∎
Theorem. (basis_finer_equiv_condition)
∀X B B' : set, basis_on X B → basis_on X B' → ((∀x ∈ X, ∀b ∈ B, x ∈ b → ∃b' ∈ B', x ∈ b' ∧ b' ⊆ b) ↔ finer_than (generated_topology X B') (generated_topology X B))
In Proofgold the corresponding term root is 9667d1... and proposition id is f5499f...
Proof:
Let X, B and B' be given.
Assume HB HB'.
Apply iffI to the current goal.
Assume Hcond.
An exact proof term for the current goal is (finer_via_basis X B B' HB HB' Hcond).
Assume Hfiner.
Let x be given.
Assume HxX.
Let b be given.
Assume HbB Hxb.
We prove the intermediate claim HbGen: b ∈ generated_topology X B.
An exact proof term for the current goal is (generated_topology_contains_basis X B HB b HbB).
We prove the intermediate claim HbGen': b ∈ generated_topology X B'.
An exact proof term for the current goal is (Hfiner b HbGen).
We prove the intermediate claim Hbprop: ∀x0 ∈ b, ∃b' ∈ B', x0 ∈ b' ∧ b' ⊆ b.
An exact proof term for the current goal is (SepE2 (𝒫 X) (λU0 : set ⇒ ∀x0 ∈ U0, ∃b0 ∈ B', x0 ∈ b0 ∧ b0 ⊆ U0) b HbGen').
An exact proof term for the current goal is (Hbprop x Hxb).
∎
Theorem. (generated_topology_finer)
∀X B T : set, basis_on X B → topology_on X T → (∀b ∈ B, b ∈ T) → finer_than T (generated_topology X B)
In Proofgold the corresponding term root is 06d86b... and proposition id is 2d0172...
Proof:
Let X, B and T be given.
Assume HBasis HT HBsub.
We prove the intermediate claim HUnionClosed: ∀Fam ∈ 𝒫 T, ⋃ Fam ∈ T.
An exact proof term for the current goal is (andER ((T ⊆ 𝒫 X ∧ Empty ∈ T) ∧ X ∈ T) (∀Fam ∈ 𝒫 T, ⋃ Fam ∈ T) (andEL (((T ⊆ 𝒫 X ∧ Empty ∈ T) ∧ X ∈ T) ∧ (∀Fam ∈ 𝒫 T, ⋃ Fam ∈ T)) (∀U ∈ T, ∀V ∈ T, U ∩ V ∈ T) HT)).
We will prove generated_topology X B ⊆ T.
Let U be given.
Assume HU.
We prove the intermediate claim HUsubX: U ⊆ X.
An exact proof term for the current goal is (PowerE X U (SepE1 (𝒫 X) (λU0 : set ⇒ ∀x0 ∈ U0, ∃b0 ∈ B, x0 ∈ b0 ∧ b0 ⊆ U0) U HU)).
We prove the intermediate claim HUprop: ∀x ∈ U, ∃b ∈ B, x ∈ b ∧ b ⊆ U.
An exact proof term for the current goal is (SepE2 (𝒫 X) (λU0 : set ⇒ ∀x0 ∈ U0, ∃b0 ∈ B, x0 ∈ b0 ∧ b0 ⊆ U0) U HU).
Set Fam to be the term {b ∈ B|b ⊆ U} of type set.
We prove the intermediate claim HFamPowB: Fam ∈ 𝒫 B.
Apply PowerI B Fam to the current goal.
Let b be given.
Assume HbFam.
An exact proof term for the current goal is (SepE1 B (λb0 : set ⇒ b0 ⊆ U) b HbFam).
We prove the intermediate claim HUnionEq: ⋃ Fam = U.
Apply set_ext to the current goal.
Let x be given.
Assume HxUnion.
Apply UnionE_impred Fam x HxUnion to the current goal.
Let b be given.
Assume Hxb HbFam.
We prove the intermediate claim HbsubU: b ⊆ U.
An exact proof term for the current goal is (SepE2 B (λb0 : set ⇒ b0 ⊆ U) b HbFam).
An exact proof term for the current goal is (HbsubU x Hxb).
Let x be given.
Assume HxU.
We prove the intermediate claim Hexb: ∃b ∈ B, x ∈ b ∧ b ⊆ U.
An exact proof term for the current goal is (HUprop x HxU).
Apply Hexb to the current goal.
Let b be given.
Assume Hbpair.
We prove the intermediate claim HbB: b ∈ B.
An exact proof term for the current goal is (andEL (b ∈ B) (x ∈ b ∧ b ⊆ U) Hbpair).
We prove the intermediate claim Hbprop: x ∈ b ∧ b ⊆ U.
An exact proof term for the current goal is (andER (b ∈ B) (x ∈ b ∧ b ⊆ U) Hbpair).
We prove the intermediate claim Hxb: x ∈ b.
An exact proof term for the current goal is (andEL (x ∈ b) (b ⊆ U) Hbprop).
We prove the intermediate claim HbsubU: b ⊆ U.
An exact proof term for the current goal is (andER (x ∈ b) (b ⊆ U) Hbprop).
We prove the intermediate claim HbT: b ∈ T.
An exact proof term for the current goal is (HBsub b HbB).
We prove the intermediate claim HbFam: b ∈ Fam.
An exact proof term for the current goal is (SepI B (λb0 : set ⇒ b0 ⊆ U) b HbB HbsubU).
An exact proof term for the current goal is (UnionI Fam x b Hxb HbFam).
We prove the intermediate claim HFamPowT: Fam ∈ 𝒫 T.
Apply PowerI T Fam to the current goal.
Let b be given.
Assume HbFam.
We prove the intermediate claim HbB: b ∈ B.
An exact proof term for the current goal is (SepE1 B (λb0 : set ⇒ b0 ⊆ U) b HbFam).
An exact proof term for the current goal is (HBsub b HbB).
We prove the intermediate claim HUnionT: ⋃ Fam ∈ T.
An exact proof term for the current goal is (HUnionClosed Fam HFamPowT).
rewrite the current goal using HUnionEq (from right to left).
An exact proof term for the current goal is HUnionT.
∎
Theorem. (topology_generated_by_basis_is_smallest)
∀X B T : set, basis_on X B → topology_on X T → (∀b ∈ B, b ∈ T) → finer_than T (generated_topology X B)
In Proofgold the corresponding term root is 06d86b... and proposition id is 2d0172...
Proof:
Let X, B and T be given.
Assume HBasis HT HBsub.
An exact proof term for the current goal is (generated_topology_finer X B T HBasis HT HBsub).
∎
Theorem. (union_of_basis_equals_open)
∀X B : set, basis_on X B → generated_topology X B = {⋃ Fam|Fam ∈ 𝒫 B}
In Proofgold the corresponding term root is 3ee001... and proposition id is 05b8bc...
Proof:
Let X and B be given.
Assume HBasis.
Apply set_ext to the current goal.
Let U be given.
Assume HU.
We prove the intermediate claim HUopen: open_in X (generated_topology X B) U.
An exact proof term for the current goal is (andI (topology_on X (generated_topology X B)) (U ∈ generated_topology X B) (lemma_topology_from_basis X B HBasis) HU).
We prove the intermediate claim HexFam: ∃Fam ∈ 𝒫 B, ⋃ Fam = U.
An exact proof term for the current goal is (open_sets_as_unions_of_basis X B HBasis U HUopen).
Apply HexFam to the current goal.
Let Fam be given.
Assume HFampair.
We prove the intermediate claim HFamPow: Fam ∈ 𝒫 B.
An exact proof term for the current goal is (andEL (Fam ∈ 𝒫 B) (⋃ Fam = U) HFampair).
We prove the intermediate claim HUnion: ⋃ Fam = U.
An exact proof term for the current goal is (andER (Fam ∈ 𝒫 B) (⋃ Fam = U) HFampair).
We prove the intermediate claim HUnionFam: ⋃ Fam ∈ {⋃ Fam0|Fam0 ∈ 𝒫 B}.
An exact proof term for the current goal is (ReplI (𝒫 B) (λFam0 : set ⇒ ⋃ Fam0) Fam HFamPow).
rewrite the current goal using HUnion (from right to left).
An exact proof term for the current goal is HUnionFam.
Let U be given.
Assume HUUnion.
We prove the intermediate claim HexFamPowRaw: ∃Fam ∈ 𝒫 B, U = ⋃ Fam.
An exact proof term for the current goal is (ReplE (𝒫 B) (λFam0 : set ⇒ ⋃ Fam0) U HUUnion).
We prove the intermediate claim HexFamPow: ∃Fam ∈ 𝒫 B, ⋃ Fam = U.
Apply HexFamPowRaw to the current goal.
Let Fam be given.
Assume HFamPair.
We prove the intermediate claim HFamPow: Fam ∈ 𝒫 B.
An exact proof term for the current goal is (andEL (Fam ∈ 𝒫 B) (U = ⋃ Fam) HFamPair).
We prove the intermediate claim HUnion: U = ⋃ Fam.
An exact proof term for the current goal is (andER (Fam ∈ 𝒫 B) (U = ⋃ Fam) HFamPair).
We use Fam to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is HFamPow.
rewrite the current goal using HUnion (from right to left).
Use reflexivity.
We prove the intermediate claim HUopen: open_in X (generated_topology X B) U.
An exact proof term for the current goal is (basis_generates_open_sets X B HBasis U HexFamPow).
An exact proof term for the current goal is (andER (topology_on X (generated_topology X B)) (U ∈ generated_topology X B) HUopen).
∎
Definition. We define singleton_basis to be λX ⇒ {{x,x}|x ∈ X} of type set → set.
In Proofgold the corresponding term root is ab4e07... and object id is 61ee14...
Theorem. (singleton_basis_is_basis)
∀X : set, basis_on X (singleton_basis X)
In Proofgold the corresponding term root is e75d9b... and proposition id is 4d1b5a...
Proof:
Let X be given.
We will prove ((singleton_basis X ⊆ 𝒫 X ∧ (∀x ∈ X, ∃b ∈ singleton_basis X, x ∈ b)) ∧ (∀b1 ∈ singleton_basis X, ∀b2 ∈ singleton_basis X, ∀x : set, x ∈ b1 → x ∈ b2 → ∃b3 ∈ singleton_basis X, x ∈ b3 ∧ b3 ⊆ b1 ∩ b2)).
Apply andI to the current goal.
Apply andI to the current goal.
We will prove singleton_basis X ⊆ 𝒫 X.
Let s be given.
Assume Hs.
Apply (ReplE_impred X (λx0 : set ⇒ {x0,x0}) s Hs) to the current goal.
Let x be given.
Assume HxX Hseq.
rewrite the current goal using Hseq (from left to right).
Apply PowerI to the current goal.
Let y be given.
Assume Hy.
Apply (UPairE y x x Hy (y ∈ X)) to the current goal.
Assume Hyx.
rewrite the current goal using Hyx (from left to right).
An exact proof term for the current goal is HxX.
Assume Hyx.
rewrite the current goal using Hyx (from left to right).
An exact proof term for the current goal is HxX.
We will prove ∀x ∈ X, ∃b ∈ singleton_basis X, x ∈ b.
Let x be given.
Assume HxX.
We use {x,x} to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is (ReplI X (λx0 : set ⇒ {x0,x0}) x HxX).
Apply UPairI1 to the current goal.
We will prove ∀b1 ∈ singleton_basis X, ∀b2 ∈ singleton_basis X, ∀x : set, x ∈ b1 → x ∈ b2 → ∃b3 ∈ singleton_basis X, x ∈ b3 ∧ b3 ⊆ b1 ∩ b2.
Let b1 be given.
Assume Hb1.
Let b2 be given.
Assume Hb2.
Let x be given.
Assume Hx1 Hx2.
Apply (ReplE_impred X (λx0 : set ⇒ {x0,x0}) b1 Hb1) to the current goal.
Let x1 be given.
Assume Hx1X Hb1eq.
Apply (ReplE_impred X (λx0 : set ⇒ {x0,x0}) b2 Hb2) to the current goal.
Let x2 be given.
Assume Hx2X Hb2eq.
We prove the intermediate claim Hx1in: x ∈ {x1,x1}.
rewrite the current goal using Hb1eq (from right to left).
An exact proof term for the current goal is Hx1.
We prove the intermediate claim Hx2in: x ∈ {x2,x2}.
rewrite the current goal using Hb2eq (from right to left).
An exact proof term for the current goal is Hx2.
We prove the intermediate claim Hx_eq_x1: x = x1.
Apply (UPairE x x1 x1 Hx1in (x = x1)) to the current goal.
Assume Hxx1.
An exact proof term for the current goal is Hxx1.
Assume Hxx1.
An exact proof term for the current goal is Hxx1.
We prove the intermediate claim Hx_eq_x2: x = x2.
Apply (UPairE x x2 x2 Hx2in (x = x2)) to the current goal.
Assume Hxx2.
An exact proof term for the current goal is Hxx2.
Assume Hxx2.
An exact proof term for the current goal is Hxx2.
We prove the intermediate claim HxX: x ∈ X.
rewrite the current goal using Hx_eq_x1 (from left to right).
An exact proof term for the current goal is Hx1X.
We use {x,x} to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is (ReplI X (λx0 : set ⇒ {x0,x0}) x HxX).
Apply andI to the current goal.
Apply UPairI1 to the current goal.
We will prove {x,x} ⊆ b1 ∩ b2.
Let y be given.
Assume Hy.
Apply (UPairE y x x Hy (y ∈ b1 ∩ b2)) to the current goal.
Assume Hyx.
rewrite the current goal using Hyx (from left to right).
Apply binintersectI to the current goal.
An exact proof term for the current goal is Hx1.
An exact proof term for the current goal is Hx2.
Assume Hyx.
rewrite the current goal using Hyx (from left to right).
Apply binintersectI to the current goal.
An exact proof term for the current goal is Hx1.
An exact proof term for the current goal is Hx2.
∎
Theorem. (generated_topology_singletons_discrete)
∀X : set, generated_topology X (singleton_basis X) = discrete_topology X
In Proofgold the corresponding term root is 047e37... and proposition id is 334958...
Proof:
Let X be given.
Apply set_ext to the current goal.
Let U be given.
Assume HUgen.
An exact proof term for the current goal is (SepE1 (𝒫 X) (λU0 : set ⇒ ∀x0 ∈ U0, ∃b ∈ singleton_basis X, x0 ∈ b ∧ b ⊆ U0) U HUgen).
Let U be given.
Assume HUinPow: U ∈ 𝒫 X.
We prove the intermediate claim HUsubX: U ⊆ X.
An exact proof term for the current goal is (PowerE X U HUinPow).
We prove the intermediate claim HUprop: ∀x ∈ U, ∃b ∈ singleton_basis X, x ∈ b ∧ b ⊆ U.
Let x be given.
Assume HxU.
We use {x,x} to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is (ReplI X (λx0 : set ⇒ {x0,x0}) x (HUsubX x HxU)).
Apply andI to the current goal.
An exact proof term for the current goal is (UPairI1 x x).
Let y be given.
Assume Hy.
Apply (UPairE y x x Hy (y ∈ U)) to the current goal.
Assume Hyx.
rewrite the current goal using Hyx (from left to right).
An exact proof term for the current goal is HxU.
Assume Hyx.
rewrite the current goal using Hyx (from left to right).
An exact proof term for the current goal is HxU.
An exact proof term for the current goal is (SepI (𝒫 X) (λU0 : set ⇒ ∀x0 ∈ U0, ∃b ∈ singleton_basis X, x0 ∈ b ∧ b ⊆ U0) U HUinPow HUprop).
∎
Definition. We define OrderedPair to be λx y ⇒ UPair x (UPair x y) of type set → set → set.
In Proofgold the corresponding term root is 1efd14... and object id is 1d4b3b...
Definition. We define R to be real of type set.
In Proofgold the corresponding term root is e26ffa... and object id is 5277ef...
Definition. We define Q to be {q ∈ R|∃p n : set, p ∈ ω ∧ n ∈ ω ∧ q ∈ R} of type set.
In Proofgold the corresponding term root is bac70e... and object id is 70b435...
Definition. We define Rlt to be λa b ⇒ a ∈ R ∧ b ∈ R ∧ a < b of type set → set → prop.
In Proofgold the corresponding term root is d991c7... and object id is ff36c7...
Definition. We define EuclidPlane to be OrderedPair R R of type set.
In Proofgold the corresponding term root is 60f7e1... and object id is d6fb09...
Definition. We define distance_R2 to be λp c ⇒ Eps_i (λr ⇒ r ∈ R) of type set → set → set.
In Proofgold the corresponding term root is dfe031... and object id is 4ce3c7...
Definition. We define circular_regions to be {U ∈ 𝒫 EuclidPlane|∃c : set, ∃r : set, c ∈ EuclidPlane ∧ r ∈ R ∧ ¬ (r = 0) ∧ U = {p ∈ EuclidPlane|Rlt (distance_R2 p c) r}} of type set.
In Proofgold the corresponding term root is 10e2a8... and object id is a0f8a5...
Definition. We define rectangular_regions to be {U ∈ 𝒫 EuclidPlane|∃a b c d : set, a ∈ R ∧ b ∈ R ∧ c ∈ R ∧ d ∈ R ∧ U = {p ∈ EuclidPlane|∃x y : set, p = OrderedPair x y ∧ Rlt a x ∧ Rlt x b ∧ Rlt c y ∧ Rlt y d}} of type set.
In Proofgold the corresponding term root is b9f57c... and object id is 094b19...
Theorem. (circular_regions_basis_plane)
basis_on EuclidPlane circular_regions
In Proofgold the corresponding term root is dc493c... and proposition id is 42b49a...
Proof:
We will prove basis_on EuclidPlane circular_regions.
The rest of this subproof is missing.
∎
Theorem. (rectangular_regions_basis_plane)
basis_on EuclidPlane rectangular_regions
In Proofgold the corresponding term root is 4ea33b... and proposition id is 747ab2...
Proof:
We will prove basis_on EuclidPlane rectangular_regions.
The rest of this subproof is missing.
∎
Theorem. (circular_rectangular_same_topology_plane)
generated_topology EuclidPlane circular_regions = generated_topology EuclidPlane rectangular_regions
In Proofgold the corresponding term root is ac3f36... and proposition id is dc1cc0...
Proof:
We will prove generated_topology EuclidPlane circular_regions = generated_topology EuclidPlane rectangular_regions.
The rest of this subproof is missing.
∎
Theorem. (lemma_finer_if_basis_refines)
∀X B B' : set, basis_on X B → basis_refines X B' (generated_topology X B) → finer_than (generated_topology X B') (generated_topology X B)
In Proofgold the corresponding term root is 92c83c... and proposition id is 6c089e...
Proof:
Let X, B and B' be given.
Assume HBasis Href.
We prove the intermediate claim Hprop: ∀U ∈ generated_topology X B, ∀x ∈ U, ∃b' ∈ B', x ∈ b' ∧ b' ⊆ U.
An exact proof term for the current goal is (andER (topology_on X (generated_topology X B)) (∀U ∈ generated_topology X B, ∀x ∈ U, ∃b' ∈ B', x ∈ b' ∧ b' ⊆ U) Href).
We will prove generated_topology X B ⊆ generated_topology X B'.
Let U be given.
Assume HU.
We prove the intermediate claim HUsubX: U ⊆ X.
An exact proof term for the current goal is (PowerE X U (SepE1 (𝒫 X) (λU0 : set ⇒ ∀x0 ∈ U0, ∃b0 ∈ B, x0 ∈ b0 ∧ b0 ⊆ U0) U HU)).
We prove the intermediate claim HUprop: ∀x ∈ U, ∃b' ∈ B', x ∈ b' ∧ b' ⊆ U.
An exact proof term for the current goal is (Hprop U HU).
An exact proof term for the current goal is (SepI (𝒫 X) (λU0 : set ⇒ ∀x0 ∈ U0, ∃b0 ∈ B', x0 ∈ b0 ∧ b0 ⊆ U0) U (PowerI X U HUsubX) HUprop).
∎
Definition. We define subbasis_on to be λX S ⇒ S ⊆ 𝒫 X of type set → set → prop.
In Proofgold the corresponding term root is 0c7446... and object id is 50a746...
Definition. We define intersection_of_family to be λFam ⇒ {x ∈ ⋃ Fam|∀U : set, U ∈ Fam → x ∈ U} of type set → set.
In Proofgold the corresponding term root is 0f045a... and object id is 1508d8...
Definition. We define finite_subcollections to be λS ⇒ {F ∈ 𝒫 S|finite F} of type set → set.
In Proofgold the corresponding term root is b6350a... and object id is b7ce2a...
Definition. We define finite_intersections_of to be λS ⇒ {intersection_of_family F|F ∈ finite_subcollections S} of type set → set.
In Proofgold the corresponding term root is c5f6f7... and object id is 0714a1...
Definition. We define basis_of_subbasis to be λ_ S ⇒ {b ∈ finite_intersections_of S|b ≠ Empty} of type set → set → set.
In Proofgold the corresponding term root is c877b8... and object id is eebbe6...
Axiom. (finite_intersection_in_basis) We take the following as an axiom:
∀X S F : set, F ∈ finite_subcollections S → intersection_of_family F ≠ Empty → intersection_of_family F ∈ basis_of_subbasis X S
In Proofgold the corresponding term root is 62df99... and proposition id is c77392...
Axiom. (subbasis_elem_in_basis) We take the following as an axiom:
∀X S s : set, s ∈ S → s ≠ Empty → s ∈ basis_of_subbasis X S
In Proofgold the corresponding term root is b04c34... and proposition id is 0edc00...
Definition. We define generated_topology_from_subbasis to be λX S ⇒ generated_topology X (basis_of_subbasis X S) of type set → set → set.
In Proofgold the corresponding term root is da6806... and object id is e91545...
Theorem. (finite_intersections_basis_of_subbasis)
∀X S : set, subbasis_on X S → basis_on X (basis_of_subbasis X S)
In Proofgold the corresponding term root is ee411a... and proposition id is b526b5...
Proof:
Let X and S be given.
Assume HS.
We will prove basis_on X (basis_of_subbasis X S).
The rest of this subproof is missing.
∎
Theorem. (topology_from_subbasis_is_topology)
∀X S : set, subbasis_on X S → topology_on X (generated_topology_from_subbasis X S)
In Proofgold the corresponding term root is 021719... and proposition id is 197c56...
Proof:
Let X and S be given.
Assume HS.
We will prove topology_on X (generated_topology_from_subbasis X S).
The rest of this subproof is missing.
∎
Theorem. (topology_generated_by_basis_is_minimal)
∀X S T : set, subbasis_on X S → topology_on X T → S ⊆ T → finer_than T (generated_topology_from_subbasis X S)
In Proofgold the corresponding term root is 5afcf8... and proposition id is a43dae...
Proof:
Let X, S and T be given.
Assume HS HT HST.
We will prove finer_than T (generated_topology_from_subbasis X S).
The rest of this subproof is missing.
∎
Theorem. (ex13_1_local_open_subset)
∀X T A : set, topology_on X T → (∀x ∈ A, ∃U ∈ T, x ∈ U ∧ U ⊆ A) → open_in X T A
In Proofgold the corresponding term root is b3939f... and proposition id is 083354...
Proof:
Let X, T and A be given.
Assume HT Hlocal.
We will prove open_in X T A.
The rest of this subproof is missing.
∎
Definition. We define a_elt to be Empty of type set.
In Proofgold the corresponding term root is e2a833... and object id is dee93d...
Definition. We define b_elt to be {Empty} of type set.
In Proofgold the corresponding term root is c95c26... and object id is 4848d8...
Definition. We define c_elt to be {{Empty}} of type set.
In Proofgold the corresponding term root is 4145a1... and object id is 07be9b...
Definition. We define abc_set to be UPair a_elt (UPair b_elt c_elt) of type set.
In Proofgold the corresponding term root is 2b7b62... and object id is 52e7a2...
Definition. We define top_abc_1 to be UPair Empty abc_set of type set.
In Proofgold the corresponding term root is 057de5... and object id is 0e20f6...
Definition. We define top_abc_2 to be 𝒫 abc_set of type set.
In Proofgold the corresponding term root is 859b09... and object id is b0da12...
Definition. We define top_abc_3 to be UPair Empty (UPair {a_elt} abc_set) of type set.
In Proofgold the corresponding term root is e6968f... and object id is f7504d...
Definition. We define top_abc_4 to be UPair Empty (UPair {b_elt} abc_set) of type set.
In Proofgold the corresponding term root is 8dde55... and object id is f500bf...
Definition. We define top_abc_5 to be UPair Empty (UPair {c_elt} abc_set) of type set.
In Proofgold the corresponding term root is 9e4589... and object id is e09d0c...
Definition. We define top_abc_6 to be UPair Empty (UPair (UPair a_elt b_elt) abc_set) of type set.
In Proofgold the corresponding term root is 521344... and object id is 098c41...
Definition. We define top_abc_7 to be UPair Empty (UPair (UPair a_elt c_elt) abc_set) of type set.
In Proofgold the corresponding term root is 2ea8af... and object id is 58bb43...
Definition. We define top_abc_8 to be UPair Empty (UPair (UPair b_elt c_elt) abc_set) of type set.
In Proofgold the corresponding term root is ff43c0... and object id is 467bb5...
Definition. We define top_abc_9 to be UPair Empty (UPair {a_elt} (UPair (UPair a_elt b_elt) abc_set)) of type set.
In Proofgold the corresponding term root is a863b1... and object id is 0307f6...
Theorem. (ex13_2_compare_nine_topologies)
topology_on abc_set top_abc_1 ∧ topology_on abc_set top_abc_2 ∧ topology_on abc_set top_abc_3 ∧ topology_on abc_set top_abc_4 ∧ topology_on abc_set top_abc_5 ∧ topology_on abc_set top_abc_6 ∧ topology_on abc_set top_abc_7 ∧ topology_on abc_set top_abc_8 ∧ topology_on abc_set top_abc_9 ∧ ∃finer_pairs : set, finer_pairs = {p ∈ 𝒫 (𝒫 (𝒫 abc_set))|∃T1 T2 : set, p = OrderedPair T1 T2 ∧ (T1 = top_abc_1 ∨ T1 = top_abc_2 ∨ T1 = top_abc_3 ∨ T1 = top_abc_4 ∨ T1 = top_abc_5 ∨ T1 = top_abc_6 ∨ T1 = top_abc_7 ∨ T1 = top_abc_8 ∨ T1 = top_abc_9) ∧ (T2 = top_abc_1 ∨ T2 = top_abc_2 ∨ T2 = top_abc_3 ∨ T2 = top_abc_4 ∨ T2 = top_abc_5 ∨ T2 = top_abc_6 ∨ T2 = top_abc_7 ∨ T2 = top_abc_8 ∨ T2 = top_abc_9) ∧ T1 ⊆ T2}
In Proofgold the corresponding term root is cb2696... and proposition id is 7cfc47...
Definition. We define Intersection_Fam to be λFam ⇒ {U ∈ 𝒫 (⋃ Fam)|∀T : set, T ∈ Fam → U ∈ T} of type set → set.
In Proofgold the corresponding term root is b73298... and object id is 980608...
Theorem. (intersection_of_family_sub_union)
∀Fam : set, intersection_of_family Fam ⊆ ⋃ Fam
In Proofgold the corresponding term root is 3fa512... and proposition id is b1db48...
Proof:
Let Fam be given.
Let x be given.
Assume Hx.
An exact proof term for the current goal is (SepE1 (⋃ Fam) (λx0 ⇒ ∀U : set, U ∈ Fam → x0 ∈ U) x Hx).
∎
Theorem. (intersection_of_family_empty)
∀Fam : set, Empty ∈ Intersection_Fam Fam
In Proofgold the corresponding term root is 2cc0b9... and proposition id is 97793e...
Proof:
Let Fam be given.
We will prove Empty ∈ Intersection_Fam Fam.
The rest of this subproof is missing.
∎
Definition. We define infinite_complement_family to be λX ⇒ {U ∈ 𝒫 X|infinite (X ∖ U) ∨ U = Empty ∨ U = X} of type set → set.
In Proofgold the corresponding term root is 7809a4... and object id is 66ca72...
Theorem. (ex13_3a_Tc_topology)
∀X : set, topology_on X (countable_complement_topology X)
In Proofgold the corresponding term root is 74c81d... and proposition id is f9a8bf...
Proof:
Let X be given.
We will prove topology_on X (countable_complement_topology X).
The rest of this subproof is missing.
∎
Theorem. (ex13_3a_countable_complement_open)
∀X : set, ∀U ∈ countable_complement_topology X, U ≠ Empty → countable (X ∖ U)
In Proofgold the corresponding term root is 2307f0... and proposition id is 850fad...
Proof:
Let X and U be given.
Assume HU: U ∈ countable_complement_topology X.
Assume Hnemp: U ≠ Empty.
We will prove countable (X ∖ U).
We prove the intermediate claim Hprop: countable (X ∖ U) ∨ U = Empty.
An exact proof term for the current goal is (SepE2 (𝒫 X) (λV : set ⇒ countable (X ∖ V) ∨ V = Empty) U HU).
We prove the intermediate claim Hcount_branch: countable (X ∖ U) → countable (X ∖ U).
Assume Hcount.
An exact proof term for the current goal is Hcount.
We prove the intermediate claim Hempty_branch: U = Empty → countable (X ∖ U).
Assume HUeq.
Apply FalseE to the current goal.
An exact proof term for the current goal is (Hnemp HUeq).
An exact proof term for the current goal is (Hprop (countable (X ∖ U)) Hcount_branch Hempty_branch).
∎
Theorem. (ex13_3a_union_helper)
∀X : set, ∀UFam ∈ 𝒫 (countable_complement_topology X), ⋃ UFam ∈ countable_complement_topology X
In Proofgold the corresponding term root is c81519... and proposition id is 6627e7...
Proof:
Let X and UFam be given.
Assume HUFam: UFam ∈ 𝒫 (countable_complement_topology X).
We will prove ⋃ UFam ∈ countable_complement_topology X.
The rest of this subproof is missing.
∎
Theorem. (ex13_3b_witness_sets)
∀X : set, ∃U V : set, U ∈ infinite_complement_family X ∧ V ∈ infinite_complement_family X ∧ ¬ (U ∩ V ∈ infinite_complement_family X)
In Proofgold the corresponding term root is 8fec1a... and proposition id is e14c12...
Proof:
Let X be given.
We will prove ∃U V : set, U ∈ infinite_complement_family X ∧ V ∈ infinite_complement_family X ∧ ¬ (U ∩ V ∈ infinite_complement_family X).
The rest of this subproof is missing.
∎
Theorem. (ex13_3b_Tinfty_not_topology)
∀X : set, ¬ topology_on X (infinite_complement_family X)
In Proofgold the corresponding term root is 43e6f4... and proposition id is 4c914e...
Proof:
Let X be given.
Assume Htop: topology_on X (infinite_complement_family X).
We will prove False.
The rest of this subproof is missing.
∎
Theorem. (ex13_3b_witness_outline)
∀X : set, ∃U V : set, U ∈ infinite_complement_family X ∧ V ∈ infinite_complement_family X
In Proofgold the corresponding term root is 9eb7de... and proposition id is 5ce8d8...
Proof:
Let X be given.
We will prove ∃U V : set, U ∈ infinite_complement_family X ∧ V ∈ infinite_complement_family X.
The rest of this subproof is missing.
∎
Theorem. (ex13_4a_intersection_topology)
∀X Fam : set, (∀T ∈ Fam, topology_on X T) → topology_on X (Intersection_Fam Fam)
In Proofgold the corresponding term root is 21ab9f... and proposition id is 9490d0...
Proof:
Let X and Fam be given.
Assume HfamTop: ∀T ∈ Fam, topology_on X T.
We will prove topology_on X (Intersection_Fam Fam).
The rest of this subproof is missing.
∎
Theorem. (ex13_4b_smallest_largest)
∀X Fam : set, ∃Tmin, topology_on X Tmin ∧ (∀T ∈ Fam, T ⊆ Tmin) ∧ (∀T', topology_on X T' ∧ (∀T ∈ Fam, T ⊆ T') → Tmin ⊆ T') ∧ ∃Tmax, topology_on X Tmax ∧ (∀T ∈ Fam, Tmax ⊆ T) ∧ (∀T', topology_on X T' ∧ (∀T ∈ Fam, T' ⊆ T) → T' ⊆ Tmax)
In Proofgold the corresponding term root is 3c2948... and proposition id is b294ae...
Proof:
Let X and Fam be given.
We will prove ∃Tmin, topology_on X Tmin ∧ (∀T ∈ Fam, T ⊆ Tmin) ∧ (∀T', topology_on X T' ∧ (∀T ∈ Fam, T ⊆ T') → Tmin ⊆ T') ∧ ∃Tmax, topology_on X Tmax ∧ (∀T ∈ Fam, Tmax ⊆ T) ∧ (∀T', topology_on X T' ∧ (∀T ∈ Fam, T' ⊆ T) → T' ⊆ Tmax).
The rest of this subproof is missing.
∎
Theorem. (ex13_4c_specific_topologies)
∃Tsmall Tall : set, topology_on abc_set Tsmall ∧ topology_on abc_set Tall
In Proofgold the corresponding term root is 5d4d94... and proposition id is b2b1df...
Proof:
We will prove ∃Tsmall Tall : set, topology_on abc_set Tsmall ∧ topology_on abc_set Tall.
The rest of this subproof is missing.
∎
Theorem. (ex13_5_basis_intersection)
∀X A : set, basis_on X A → generated_topology X A = Intersection_Fam {T ∈ 𝒫 (𝒫 X)|topology_on X T ∧ A ⊆ T}
In Proofgold the corresponding term root is 74dce9... and proposition id is 5c20d0...
Proof:
Let X and A be given.
Assume HA: basis_on X A.
We will prove generated_topology X A = Intersection_Fam {T ∈ 𝒫 (𝒫 X)|topology_on X T ∧ A ⊆ T}.
The rest of this subproof is missing.
∎
Definition. We define rational_numbers to be ω of type set.
In Proofgold the corresponding term root is 6fc30a... and object id is ad9ad8...
Definition. We define open_interval to be λa b ⇒ {x ∈ R|Rlt a x ∧ Rlt x b} of type set → set → set.
In Proofgold the corresponding term root is 58cccb... and object id is 7f9904...
Definition. We define halfopen_interval_left to be λa b ⇒ {x ∈ R|Rlt a x ∧ ¬ (Rlt b x)} of type set → set → set.
In Proofgold the corresponding term root is 94dc48... and object id is 6ac391...
Definition. We define R_standard_basis to be ⋃a ∈ R{open_interval a b|b ∈ R} of type set.
In Proofgold the corresponding term root is 6c64ea... and object id is e0fd50...
Definition. We define R_standard_topology to be generated_topology R R_standard_basis of type set.
In Proofgold the corresponding term root is f35291... and object id is 9f1b53...
Definition. We define R_lower_limit_basis to be ⋃a ∈ R{halfopen_interval_left a b|b ∈ R} of type set.
In Proofgold the corresponding term root is 819f82... and object id is f6ef78...
Definition. We define R_lower_limit_topology to be generated_topology R R_lower_limit_basis of type set.
In Proofgold the corresponding term root is d07fcd... and object id is 6262c6...
Definition. We define inv_nat to be λn ⇒ n of type set → set.
In Proofgold the corresponding term root is c66d8b... and object id is ede5a9...
Axiom. (inv_nat_real) We take the following as an axiom:
∀n : set, n ∈ ω → inv_nat n ∈ R
In Proofgold the corresponding term root is 7ec3d3... and proposition id is 81c77a...
Definition. We define K_set to be {inv_nat n|n ∈ ω} of type set.
In Proofgold the corresponding term root is 000761... and object id is 8f3cdc...
Definition. We define R_K_basis to be ⋃a ∈ R{open_interval a b ∖ K_set|b ∈ R} of type set.
In Proofgold the corresponding term root is 7506b3... and object id is 52e9b9...
Definition. We define R_K_topology to be generated_topology R (R_standard_basis ∪ R_K_basis) of type set.
In Proofgold the corresponding term root is 8c6684... and object id is 1a239c...
Theorem. (ex13_6_Rl_RK_not_comparable)
¬ finer_than R_lower_limit_topology R_K_topology ∧ ¬ finer_than R_K_topology R_lower_limit_topology
In Proofgold the corresponding term root is 0c3a84... and proposition id is 14de39...
Proof:
We will prove ¬ finer_than R_lower_limit_topology R_K_topology ∧ ¬ finer_than R_K_topology R_lower_limit_topology.
The rest of this subproof is missing.
∎
Definition. We define R_finite_complement_topology to be countable_complement_topology R of type set.
In Proofgold the corresponding term root is 88dd86... and object id is 59d229...
Definition. We define R_upper_limit_topology to be R_lower_limit_topology of type set.
In Proofgold the corresponding term root is d07fcd... and object id is 6262c6...
Definition. We define R_ray_topology to be {U ∈ 𝒫 R|U = Empty ∨ U = R ∨ (∃a ∈ R, {x ∈ R|Rlt a x} ⊆ U)} of type set.
In Proofgold the corresponding term root is 226ebb... and object id is cb88aa...
Theorem. (ex13_7_R_topology_containments)
finer_than R_upper_limit_topology R_standard_topology ∧ finer_than R_K_topology R_standard_topology ∧ finer_than R_standard_topology R_finite_complement_topology ∧ finer_than R_standard_topology R_ray_topology
In Proofgold the corresponding term root is 92a8be... and proposition id is cdce80...
Definition. We define rational_open_intervals_basis to be ⋃q1 ∈ rational_numbers{open_interval q1 q2|q2 ∈ rational_numbers} of type set.
In Proofgold the corresponding term root is 386f57... and object id is 768847...
Theorem. (ex13_8a_rational_intervals_basis_standard)
basis_on R rational_open_intervals_basis ∧ generated_topology R rational_open_intervals_basis = R_standard_topology
In Proofgold the corresponding term root is e0d4bb... and proposition id is d09c7c...
Definition. We define rational_halfopen_intervals_basis to be ⋃q1 ∈ rational_numbers{halfopen_interval_left q1 q2|q2 ∈ rational_numbers} of type set.
In Proofgold the corresponding term root is 22185c... and object id is 3e9ee6...
Theorem. (ex13_8b_halfopen_rational_basis_topology)
basis_on R rational_halfopen_intervals_basis ∧ generated_topology R rational_halfopen_intervals_basis ≠ R_lower_limit_topology
In Proofgold the corresponding term root is d1adc9... and proposition id is 24c6ca...
Definition. We define order_rel to be λX a b ⇒ (X = R ∧ Rlt a b) ∨ (X = OrderedPair R R ∧ ∃a1 a2 b1 b2 : set, a = OrderedPair a1 a2 ∧ b = OrderedPair b1 b2 ∧ (Rlt a1 b1 ∨ (a1 = b1 ∧ Rlt a2 b2))) of type set → set → set → prop.
In Proofgold the corresponding term root is f6d930... and object id is 59c307...
Definition. We define order_topology_basis to be λX ⇒ ({I ∈ 𝒫 X|∃a ∈ X, ∃b ∈ X, I = {x ∈ X|order_rel X a x ∧ order_rel X x b}} ∪ {I ∈ 𝒫 X|∃b ∈ X, I = {x ∈ X|order_rel X x b}} ∪ {I ∈ 𝒫 X|∃a ∈ X, I = {x ∈ X|order_rel X a x}}) of type set → set.
In Proofgold the corresponding term root is 43f6dc... and object id is 010605...
Definition. We define order_topology to be λX ⇒ generated_topology X (order_topology_basis X) of type set → set.
In Proofgold the corresponding term root is 55c6b4... and object id is d91bea...
Theorem. (order_topology_is_topology)
∀X : set, topology_on X (order_topology X)
In Proofgold the corresponding term root is 426b84... and proposition id is 4a124a...
Proof:
Let X be given.
We will prove topology_on X (order_topology X).
The rest of this subproof is missing.
∎
Definition. We define open_ray_upper to be λX a ⇒ {x ∈ X|order_rel X a x} of type set → set → set.
In Proofgold the corresponding term root is dd5e1a... and object id is 6eeff7...
Definition. We define open_ray_lower to be λX a ⇒ {x ∈ X|order_rel X x a} of type set → set → set.
In Proofgold the corresponding term root is 2714e6... and object id is c8abbf...
Theorem. (open_rays_subbasis_for_order_topology)
∀X : set, ∃S : set, generated_topology X S = order_topology X
In Proofgold the corresponding term root is 8a2bdb... and proposition id is a30576...
Proof:
Let X be given.
We will prove ∃S : set, generated_topology X S = order_topology X.
The rest of this subproof is missing.
∎
Theorem. (standard_topology_is_order_topology)
order_topology R = R_standard_topology
In Proofgold the corresponding term root is a52ecd... and proposition id is 787b32...
Proof:
We will prove order_topology R = R_standard_topology.
The rest of this subproof is missing.
∎
Definition. We define R2_dictionary_order_topology to be order_topology (OrderedPair R R) of type set.
In Proofgold the corresponding term root is 9ce474... and object id is eb7cbd...
Theorem. (dictionary_order_topology_is_topology)
topology_on (OrderedPair R R) R2_dictionary_order_topology
In Proofgold the corresponding term root is d37574... and proposition id is d3b78e...
Proof:
We will prove topology_on (OrderedPair R R) R2_dictionary_order_topology.
The rest of this subproof is missing.
∎
Theorem. (rectangles_basis_for_R2)
∃B : set, basis_on (OrderedPair R R) B ∧ generated_topology (OrderedPair R R) B = R2_dictionary_order_topology
In Proofgold the corresponding term root is cb5ea4... and proposition id is 2bc3e3...
Proof:
We will prove ∃B : set, basis_on (OrderedPair R R) B ∧ generated_topology (OrderedPair R R) B = R2_dictionary_order_topology.
The rest of this subproof is missing.
∎
Definition. We define Zplus to be ω of type set.
In Proofgold the corresponding term root is 6fc30a... and object id is ad9ad8...
Theorem. (order_topology_on_Zplus_discrete)
order_topology Zplus = discrete_topology Zplus
In Proofgold the corresponding term root is adf664... and proposition id is d27db7...
Proof:
We will prove order_topology Zplus = discrete_topology Zplus.
The rest of this subproof is missing.
∎
Definition. We define two_by_nat to be OrderedPair 2 ω of type set.
In Proofgold the corresponding term root is d20d45... and object id is e68c21...
Definition. We define two_by_nat_order_topology to be order_topology two_by_nat of type set.
In Proofgold the corresponding term root is b14da5... and object id is 5d1ec8...
Theorem. (two_by_nat_not_discrete)
¬ (two_by_nat_order_topology = discrete_topology two_by_nat)
In Proofgold the corresponding term root is 76d3f6... and proposition id is a3295e...
Proof:
We will prove ¬ (two_by_nat_order_topology = discrete_topology two_by_nat).
The rest of this subproof is missing.
∎
Definition. We define rectangle_set to be λU V ⇒ OrderedPair U V of type set → set → set.
In Proofgold the corresponding term root is 1efd14... and object id is 1d4b3b...
Definition. We define product_subbasis to be λX Tx Y Ty ⇒ ⋃U ∈ Tx{rectangle_set U V|V ∈ Ty} of type set → set → set → set → set.
In Proofgold the corresponding term root is d000f5... and object id is 3243a3...
Definition. We define product_topology to be λX Tx Y Ty ⇒ generated_topology (OrderedPair X Y) (product_subbasis X Tx Y Ty) of type set → set → set → set → set.
In Proofgold the corresponding term root is 2412fa... and object id is 172edd...
Theorem. (product_topology_is_topology)
∀X Tx Y Ty : set, topology_on X Tx → topology_on Y Ty → topology_on (OrderedPair X Y) (product_topology X Tx Y Ty)
In Proofgold the corresponding term root is 32e092... and proposition id is c0efeb...
Proof:
Let X, Tx, Y and Ty be given.
Assume HTx: topology_on X Tx.
Assume HTy: topology_on Y Ty.
We will prove topology_on (OrderedPair X Y) (product_topology X Tx Y Ty).
The rest of this subproof is missing.
∎
Theorem. (product_basis_generates)
∀X Tx Y Ty Bx By : set, basis_on X Bx ∧ generated_topology X Bx = Tx → basis_on Y By ∧ generated_topology Y By = Ty → ∃B : set, basis_on (OrderedPair X Y) B ∧ (∀U ∈ Bx, ∀V ∈ By, OrderedPair U V ∈ B) ∧ generated_topology (OrderedPair X Y) B = product_topology X Tx Y Ty
In Proofgold the corresponding term root is db6d52... and proposition id is 019159...
Proof:
Let X, Tx, Y, Ty, Bx and By be given.
Assume HBx: basis_on X Bx ∧ generated_topology X Bx = Tx.
Assume HBy: basis_on Y By ∧ generated_topology Y By = Ty.
We will prove ∃B : set, basis_on (OrderedPair X Y) B ∧ (∀U ∈ Bx, ∀V ∈ By, OrderedPair U V ∈ B) ∧ generated_topology (OrderedPair X Y) B = product_topology X Tx Y Ty.
The rest of this subproof is missing.
∎
Definition. We define projection1 to be λX Y ⇒ {p ∈ 𝒫 (OrderedPair (OrderedPair X Y) X)|∃x : set, ∃y : set, x ∈ X ∧ y ∈ Y ∧ p = UPair (OrderedPair x y) x} of type set → set → set.
In Proofgold the corresponding term root is ca91af... and object id is 8a192b...
Definition. We define projection2 to be λX Y ⇒ {p ∈ 𝒫 (OrderedPair (OrderedPair X Y) Y)|∃x : set, ∃y : set, x ∈ X ∧ y ∈ Y ∧ p = UPair (OrderedPair x y) y} of type set → set → set.
In Proofgold the corresponding term root is 14f7fe... and object id is 8bf188...
Theorem. (product_subbasis_from_projections)
∀X Tx Y Ty : set, topology_on X Tx → topology_on Y Ty → ∃S : set, S = product_subbasis X Tx Y Ty ∧ generated_topology (OrderedPair X Y) S = product_topology X Tx Y Ty
In Proofgold the corresponding term root is 253a67... and proposition id is 4d57ea...
Proof:
Let X, Tx, Y and Ty be given.
Assume HTx: topology_on X Tx.
Assume HTy: topology_on Y Ty.
We will prove ∃S : set, S = product_subbasis X Tx Y Ty ∧ generated_topology (OrderedPair X Y) S = product_topology X Tx Y Ty.
The rest of this subproof is missing.
∎
Definition. We define apply_fun to be λf x ⇒ Eps_i (λy ⇒ UPair x y ∈ f) of type set → set → set.
In Proofgold the corresponding term root is f1f5ba... and object id is 0202ec...
Definition. We define function_on to be λf X Y ⇒ ∀x : set, x ∈ X → apply_fun f x ∈ Y of type set → set → set → prop.
In Proofgold the corresponding term root is 2b38fd... and object id is 7f6746...
Definition. We define function_space to be λX Y ⇒ {f ∈ 𝒫 (OrderedPair X Y)|function_on f X Y} of type set → set → set.
In Proofgold the corresponding term root is 6f02e1... and object id is 287ca0...
Definition. We define const_family to be λI X ⇒ {UPair i X|i ∈ I} of type set → set → set.
In Proofgold the corresponding term root is 277423... and object id is 39b0e1...
Definition. We define product_component to be λXi i ⇒ apply_fun Xi i of type set → set → set.
In Proofgold the corresponding term root is f1f5ba... and object id is 0202ec...
Definition. We define product_component_topology to be λXi i ⇒ apply_fun Xi i of type set → set → set.
In Proofgold the corresponding term root is f1f5ba... and object id is 0202ec...
Definition. We define product_space to be λI Xi ⇒ {f ∈ 𝒫 (⋃ Xi)|function_on f I (⋃ Xi) ∧ ∀i : set, i ∈ I → apply_fun f i ∈ apply_fun Xi i} of type set → set → set.
In Proofgold the corresponding term root is b17a06... and object id is 853a98...
Definition. We define product_cylinder to be λI Xi i U ⇒ {f ∈ product_space I Xi|i ∈ I ∧ U ∈ apply_fun Xi i ∧ apply_fun f i ∈ U} of type set → set → set → set → set.
In Proofgold the corresponding term root is ae3888... and object id is 812faf...
Definition. We define product_subbasis_full to be λI Xi ⇒ ⋃i ∈ I{product_cylinder I Xi i U|U ∈ apply_fun Xi i} of type set → set → set.
In Proofgold the corresponding term root is 2d8dcd... and object id is a1e49a...
Definition. We define product_topology_full to be λI Xi ⇒ generated_topology (product_space I Xi) (product_subbasis_full I Xi) of type set → set → set.
In Proofgold the corresponding term root is 48f036... and object id is 7ce851...
Definition. We define box_topology to be λI Xi ⇒ generated_topology (product_space I Xi) (𝒫 (product_space I Xi)) of type set → set → set.
In Proofgold the corresponding term root is f6133f... and object id is 6725df...
Definition. We define countable_product_space to be λI Xi ⇒ product_space I Xi of type set → set → set.
In Proofgold the corresponding term root is b17a06... and object id is 853a98...
Definition. We define countable_product_topology to be λI Xi ⇒ product_topology_full I Xi of type set → set → set.
In Proofgold the corresponding term root is 48f036... and object id is 7ce851...
Definition. We define euclidean_space to be λn ⇒ product_space n (const_family n R) of type set → set.
In Proofgold the corresponding term root is e95b4f... and object id is 0693de...
Definition. We define euclidean_topology to be λn ⇒ product_topology_full n (const_family n R) of type set → set.
In Proofgold the corresponding term root is dea5c5... and object id is 60da0f...
Definition. We define R2_standard_topology to be product_topology R R_standard_topology R R_standard_topology of type set.
In Proofgold the corresponding term root is e34d9f... and object id is 2642d2...
Theorem. (R2_standard_equals_product)
R2_standard_topology = product_topology R R_standard_topology R R_standard_topology
In Proofgold the corresponding term root is 177c56... and proposition id is b2c79b...
Proof:
We will prove R2_standard_topology = product_topology R R_standard_topology R R_standard_topology.
The rest of this subproof is missing.
∎
Definition. We define subspace_topology to be λX Tx Y ⇒ {U ∈ 𝒫 Y|∃V ∈ Tx, U = V ∩ Y} of type set → set → set → set.
In Proofgold the corresponding term root is f2cf4f... and object id is 09e9a9...
Theorem. (subspace_topology_is_topology)
∀X Tx Y : set, topology_on X Tx → Y ⊆ X → topology_on Y (subspace_topology X Tx Y)
In Proofgold the corresponding term root is ead4cf... and proposition id is 957991...
Proof:
Let X, Tx and Y be given.
Assume HTx: topology_on X Tx.
Assume HY: Y ⊆ X.
We will prove topology_on Y (subspace_topology X Tx Y).
We will prove subspace_topology X Tx Y ⊆ 𝒫 Y ∧ Empty ∈ subspace_topology X Tx Y ∧ Y ∈ subspace_topology X Tx Y ∧ (∀UFam ∈ 𝒫 (subspace_topology X Tx Y), ⋃ UFam ∈ subspace_topology X Tx Y) ∧ (∀U ∈ subspace_topology X Tx Y, ∀V ∈ subspace_topology X Tx Y, U ∩ V ∈ subspace_topology X Tx Y).
Apply andI to the current goal.
We will prove (subspace_topology X Tx Y ⊆ 𝒫 Y ∧ Empty ∈ subspace_topology X Tx Y) ∧ Y ∈ subspace_topology X Tx Y ∧ (∀UFam ∈ 𝒫 (subspace_topology X Tx Y), ⋃ UFam ∈ subspace_topology X Tx Y).
Apply andI to the current goal.
We will prove subspace_topology X Tx Y ⊆ 𝒫 Y ∧ Empty ∈ subspace_topology X Tx Y ∧ Y ∈ subspace_topology X Tx Y.
Apply andI to the current goal.
We will prove subspace_topology X Tx Y ⊆ 𝒫 Y ∧ Empty ∈ subspace_topology X Tx Y.
Apply andI to the current goal.
We will prove subspace_topology X Tx Y ⊆ 𝒫 Y.
Let U be given.
Assume HU: U ∈ subspace_topology X Tx Y.
An exact proof term for the current goal is (SepE1 (𝒫 Y) (λU0 : set ⇒ ∃V ∈ Tx, U0 = V ∩ Y) U HU).
We will prove Empty ∈ subspace_topology X Tx Y.
We prove the intermediate claim HEmptyTx: Empty ∈ Tx.
An exact proof term for the current goal is (topology_has_empty X Tx HTx).
We prove the intermediate claim HPred: ∃V ∈ Tx, Empty = V ∩ Y.
We use Empty to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is HEmptyTx.
We will prove Empty = Empty ∩ Y.
We prove the intermediate claim H1: Empty ∩ Y = Empty.
Apply Empty_Subq_eq to the current goal.
An exact proof term for the current goal is (binintersect_Subq_1 Empty Y).
rewrite the current goal using H1 (from left to right).
Use reflexivity.
An exact proof term for the current goal is (SepI (𝒫 Y) (λU0 : set ⇒ ∃V ∈ Tx, U0 = V ∩ Y) Empty (Empty_In_Power Y) HPred).
We will prove Y ∈ subspace_topology X Tx Y.
We prove the intermediate claim HXTx: X ∈ Tx.
An exact proof term for the current goal is (topology_has_X X Tx HTx).
We prove the intermediate claim HPredY: ∃V ∈ Tx, Y = V ∩ Y.
We use X to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is HXTx.
We will prove Y = X ∩ Y.
Apply set_ext to the current goal.
Let y be given.
Assume Hy: y ∈ Y.
Apply binintersectI to the current goal.
An exact proof term for the current goal is (HY y Hy).
An exact proof term for the current goal is Hy.
An exact proof term for the current goal is (binintersect_Subq_2 X Y).
An exact proof term for the current goal is (SepI (𝒫 Y) (λU0 : set ⇒ ∃V ∈ Tx, U0 = V ∩ Y) Y (Self_In_Power Y) HPredY).
We will prove ∀UFam ∈ 𝒫 (subspace_topology X Tx Y), ⋃ UFam ∈ subspace_topology X Tx Y.
Let UFam be given.
Assume HUFam: UFam ∈ 𝒫 (subspace_topology X Tx Y).
We will prove ⋃ UFam ∈ subspace_topology X Tx Y.
We prove the intermediate claim HUFamsub: UFam ⊆ subspace_topology X Tx Y.
An exact proof term for the current goal is (PowerE (subspace_topology X Tx Y) UFam HUFam).
Set VFam to be the term {V ∈ Tx|∃U ∈ UFam, U = V ∩ Y}.
We prove the intermediate claim HVFamTx: VFam ⊆ Tx.
Let V be given.
Assume HV: V ∈ VFam.
An exact proof term for the current goal is (SepE1 Tx (λV0 ⇒ ∃U ∈ UFam, U = V0 ∩ Y) V HV).
We prove the intermediate claim HVFamPower: VFam ∈ 𝒫 Tx.
Apply PowerI to the current goal.
An exact proof term for the current goal is HVFamTx.
We prove the intermediate claim HUnionVFam: ⋃ VFam ∈ Tx.
An exact proof term for the current goal is (topology_union_closed X Tx VFam HTx HVFamTx).
We prove the intermediate claim HUnionEq: ⋃ UFam = (⋃ VFam) ∩ Y.
Apply set_ext to the current goal.
Let x be given.
Assume Hx: x ∈ ⋃ UFam.
Apply UnionE_impred UFam x Hx to the current goal.
Let U be given.
Assume HxU: x ∈ U.
Assume HUinFam: U ∈ UFam.
We prove the intermediate claim HUinSubspace: U ∈ subspace_topology X Tx Y.
An exact proof term for the current goal is (HUFamsub U HUinFam).
We prove the intermediate claim HUexists: ∃V ∈ Tx, U = V ∩ Y.
An exact proof term for the current goal is (SepE2 (𝒫 Y) (λU0 : set ⇒ ∃V ∈ Tx, U0 = V ∩ Y) U HUinSubspace).
Apply HUexists to the current goal.
Let V be given.
Assume HVandEq.
Apply HVandEq to the current goal.
Assume HVTx: V ∈ Tx.
Assume HUeq: U = V ∩ Y.
Apply binintersectI to the current goal.
We will prove x ∈ ⋃ VFam.
We prove the intermediate claim HxV: x ∈ V.
We prove the intermediate claim HxVY: x ∈ V ∩ Y.
rewrite the current goal using HUeq (from right to left).
An exact proof term for the current goal is HxU.
An exact proof term for the current goal is (binintersectE1 V Y x HxVY).
We prove the intermediate claim HVinVFam: V ∈ VFam.
Apply (SepI Tx (λV0 ⇒ ∃U ∈ UFam, U = V0 ∩ Y) V HVTx) to the current goal.
We use U to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is HUinFam.
An exact proof term for the current goal is HUeq.
An exact proof term for the current goal is (UnionI VFam x V HxV HVinVFam).
We will prove x ∈ Y.
We prove the intermediate claim HxVY: x ∈ V ∩ Y.
rewrite the current goal using HUeq (from right to left).
An exact proof term for the current goal is HxU.
An exact proof term for the current goal is (binintersectE2 V Y x HxVY).
Let x be given.
Assume Hx: x ∈ (⋃ VFam) ∩ Y.
We prove the intermediate claim HxUnionV: x ∈ ⋃ VFam.
An exact proof term for the current goal is (binintersectE1 (⋃ VFam) Y x Hx).
We prove the intermediate claim HxY: x ∈ Y.
An exact proof term for the current goal is (binintersectE2 (⋃ VFam) Y x Hx).
Apply UnionE_impred VFam x HxUnionV to the current goal.
Let V be given.
Assume HxV: x ∈ V.
Assume HVinVFam: V ∈ VFam.
We prove the intermediate claim HVexists: ∃U ∈ UFam, U = V ∩ Y.
An exact proof term for the current goal is (SepE2 Tx (λV0 ⇒ ∃U ∈ UFam, U = V0 ∩ Y) V HVinVFam).
Apply HVexists to the current goal.
Let U be given.
Assume HUandEq.
Apply HUandEq to the current goal.
Assume HUinFam: U ∈ UFam.
Assume HUeq: U = V ∩ Y.
We prove the intermediate claim HxU: x ∈ U.
rewrite the current goal using HUeq (from left to right).
Apply binintersectI to the current goal.
An exact proof term for the current goal is HxV.
An exact proof term for the current goal is HxY.
An exact proof term for the current goal is (UnionI UFam x U HxU HUinFam).
We prove the intermediate claim HPredUnion: ∃V ∈ Tx, ⋃ UFam = V ∩ Y.
We use (⋃ VFam) to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is HUnionVFam.
An exact proof term for the current goal is HUnionEq.
We prove the intermediate claim HUnionInPowerY: ⋃ UFam ∈ 𝒫 Y.
Apply PowerI to the current goal.
Let x be given.
Assume Hx: x ∈ ⋃ UFam.
We prove the intermediate claim HxVY: x ∈ (⋃ VFam) ∩ Y.
rewrite the current goal using HUnionEq (from right to left).
An exact proof term for the current goal is Hx.
An exact proof term for the current goal is (binintersectE2 (⋃ VFam) Y x HxVY).
An exact proof term for the current goal is (SepI (𝒫 Y) (λU0 : set ⇒ ∃V ∈ Tx, U0 = V ∩ Y) (⋃ UFam) HUnionInPowerY HPredUnion).
We will prove ∀U ∈ subspace_topology X Tx Y, ∀V ∈ subspace_topology X Tx Y, U ∩ V ∈ subspace_topology X Tx Y.
Let U be given.
Assume HU: U ∈ subspace_topology X Tx Y.
Let V be given.
Assume HV: V ∈ subspace_topology X Tx Y.
We will prove U ∩ V ∈ subspace_topology X Tx Y.
We prove the intermediate claim HUexists: ∃V1 ∈ Tx, U = V1 ∩ Y.
An exact proof term for the current goal is (SepE2 (𝒫 Y) (λU0 : set ⇒ ∃V ∈ Tx, U0 = V ∩ Y) U HU).
We prove the intermediate claim HVexists: ∃V2 ∈ Tx, V = V2 ∩ Y.
An exact proof term for the current goal is (SepE2 (𝒫 Y) (λV0 : set ⇒ ∃V ∈ Tx, V0 = V ∩ Y) V HV).
Apply HUexists to the current goal.
Let V1 be given.
Assume HV1andEq.
Apply HV1andEq to the current goal.
Assume HV1Tx: V1 ∈ Tx.
Assume HUeq: U = V1 ∩ Y.
Apply HVexists to the current goal.
Let V2 be given.
Assume HV2andEq.
Apply HV2andEq to the current goal.
Assume HV2Tx: V2 ∈ Tx.
Assume HVeq: V = V2 ∩ Y.
We prove the intermediate claim HV1V2: V1 ∩ V2 ∈ Tx.
An exact proof term for the current goal is (topology_binintersect_closed X Tx V1 V2 HTx HV1Tx HV2Tx).
We prove the intermediate claim HIntEq: U ∩ V = (V1 ∩ V2) ∩ Y.
rewrite the current goal using HUeq (from left to right).
rewrite the current goal using HVeq (from left to right).
We will prove (V1 ∩ Y) ∩ (V2 ∩ Y) = (V1 ∩ V2) ∩ Y.
Apply set_ext to the current goal.
Let x be given.
Assume Hx: x ∈ (V1 ∩ Y) ∩ (V2 ∩ Y).
We prove the intermediate claim HxV1Y: x ∈ V1 ∩ Y.
An exact proof term for the current goal is (binintersectE1 (V1 ∩ Y) (V2 ∩ Y) x Hx).
We prove the intermediate claim HxV2Y: x ∈ V2 ∩ Y.
An exact proof term for the current goal is (binintersectE2 (V1 ∩ Y) (V2 ∩ Y) x Hx).
We prove the intermediate claim HxV1: x ∈ V1.
An exact proof term for the current goal is (binintersectE1 V1 Y x HxV1Y).
We prove the intermediate claim HxV2: x ∈ V2.
An exact proof term for the current goal is (binintersectE1 V2 Y x HxV2Y).
We prove the intermediate claim HxY: x ∈ Y.
An exact proof term for the current goal is (binintersectE2 V1 Y x HxV1Y).
Apply binintersectI to the current goal.
Apply binintersectI to the current goal.
An exact proof term for the current goal is HxV1.
An exact proof term for the current goal is HxV2.
An exact proof term for the current goal is HxY.
Let x be given.
Assume Hx: x ∈ (V1 ∩ V2) ∩ Y.
We prove the intermediate claim HxV1V2: x ∈ V1 ∩ V2.
An exact proof term for the current goal is (binintersectE1 (V1 ∩ V2) Y x Hx).
We prove the intermediate claim HxY: x ∈ Y.
An exact proof term for the current goal is (binintersectE2 (V1 ∩ V2) Y x Hx).
We prove the intermediate claim HxV1: x ∈ V1.
An exact proof term for the current goal is (binintersectE1 V1 V2 x HxV1V2).
We prove the intermediate claim HxV2: x ∈ V2.
An exact proof term for the current goal is (binintersectE2 V1 V2 x HxV1V2).
Apply binintersectI to the current goal.
Apply binintersectI to the current goal.
An exact proof term for the current goal is HxV1.
An exact proof term for the current goal is HxY.
Apply binintersectI to the current goal.
An exact proof term for the current goal is HxV2.
An exact proof term for the current goal is HxY.
We prove the intermediate claim HPredInt: ∃W ∈ Tx, U ∩ V = W ∩ Y.
We use (V1 ∩ V2) to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is HV1V2.
An exact proof term for the current goal is HIntEq.
We prove the intermediate claim HIntInPowerY: U ∩ V ∈ 𝒫 Y.
Apply PowerI to the current goal.
Let x be given.
Assume Hx: x ∈ U ∩ V.
We prove the intermediate claim HxU: x ∈ U.
An exact proof term for the current goal is (binintersectE1 U V x Hx).
We prove the intermediate claim HUinPowerY: U ∈ 𝒫 Y.
An exact proof term for the current goal is (SepE1 (𝒫 Y) (λU0 : set ⇒ ∃V ∈ Tx, U0 = V ∩ Y) U HU).
We prove the intermediate claim HUsub: U ⊆ Y.
An exact proof term for the current goal is (PowerE Y U HUinPowerY).
An exact proof term for the current goal is (HUsub x HxU).
An exact proof term for the current goal is (SepI (𝒫 Y) (λU0 : set ⇒ ∃V ∈ Tx, U0 = V ∩ Y) (U ∩ V) HIntInPowerY HPredInt).
∎
Theorem. (open_in_subspace_iff)
∀X Tx Y U : set, topology_on X Tx → Y ⊆ X → U ⊆ Y → (open_in Y (subspace_topology X Tx Y) U ↔ ∃V ∈ Tx, U = V ∩ Y)
In Proofgold the corresponding term root is bce457... and proposition id is c2f7fe...
Proof:
Let X, Tx, Y and U be given.
Assume HTx: topology_on X Tx.
Assume HY: Y ⊆ X.
Assume HU: U ⊆ Y.
We will prove open_in Y (subspace_topology X Tx Y) U ↔ ∃V ∈ Tx, U = V ∩ Y.
Apply iffI to the current goal.
Assume HopenU: open_in Y (subspace_topology X Tx Y) U.
We will prove ∃V ∈ Tx, U = V ∩ Y.
We prove the intermediate claim HUinSubspace: U ∈ subspace_topology X Tx Y.
An exact proof term for the current goal is (andER (topology_on Y (subspace_topology X Tx Y)) (U ∈ subspace_topology X Tx Y) HopenU).
An exact proof term for the current goal is (SepE2 (𝒫 Y) (λU0 : set ⇒ ∃V ∈ Tx, U0 = V ∩ Y) U HUinSubspace).
Assume Hexists: ∃V ∈ Tx, U = V ∩ Y.
We will prove open_in Y (subspace_topology X Tx Y) U.
We will prove topology_on Y (subspace_topology X Tx Y) ∧ U ∈ subspace_topology X Tx Y.
Apply andI to the current goal.
We will prove topology_on Y (subspace_topology X Tx Y).
An exact proof term for the current goal is (subspace_topology_is_topology X Tx Y HTx HY).
We will prove U ∈ subspace_topology X Tx Y.
We prove the intermediate claim HUinPowerY: U ∈ 𝒫 Y.
Apply PowerI to the current goal.
An exact proof term for the current goal is HU.
An exact proof term for the current goal is (SepI (𝒫 Y) (λU0 : set ⇒ ∃V ∈ Tx, U0 = V ∩ Y) U HUinPowerY Hexists).
∎
Theorem. (subspace_basis)
∀X Tx Y B : set, topology_on X Tx → basis_on X B ∧ generated_topology X B = Tx → basis_on Y {b ∈ B|b ⊆ Y} ∧ generated_topology Y {b ∈ B|b ⊆ Y} = subspace_topology X Tx Y
In Proofgold the corresponding term root is a94bd0... and proposition id is e84164...
Proof:
Let X, Tx, Y and B be given.
Assume HTx: topology_on X Tx.
Assume HB: basis_on X B ∧ generated_topology X B = Tx.
We will prove basis_on Y {b ∈ B|b ⊆ Y} ∧ generated_topology Y {b ∈ B|b ⊆ Y} = subspace_topology X Tx Y.
The rest of this subproof is missing.
∎
Theorem. (open_in_subspace_if_ambient_open)
∀X Tx Y U : set, topology_on X Tx → Y ∈ Tx → U ⊆ Y → open_in Y (subspace_topology X Tx Y) U → U ∈ Tx
In Proofgold the corresponding term root is 9cf0dc... and proposition id is 941699...
Proof:
Let X, Tx, Y and U be given.
Assume HTx: topology_on X Tx.
Assume HY: Y ∈ Tx.
Assume HU: U ⊆ Y.
Assume HUopen: open_in Y (subspace_topology X Tx Y) U.
We will prove U ∈ Tx.
We prove the intermediate claim HYsub: Y ⊆ X.
An exact proof term for the current goal is (topology_elem_subset X Tx Y HTx HY).
We prove the intermediate claim HUiffExists: open_in Y (subspace_topology X Tx Y) U ↔ ∃V ∈ Tx, U = V ∩ Y.
An exact proof term for the current goal is (open_in_subspace_iff X Tx Y U HTx HYsub HU).
We prove the intermediate claim Hexists: ∃V ∈ Tx, U = V ∩ Y.
An exact proof term for the current goal is (iffEL (open_in Y (subspace_topology X Tx Y) U) (∃V ∈ Tx, U = V ∩ Y) HUiffExists HUopen).
Apply Hexists to the current goal.
Let V be given.
Assume HVandEq.
Apply HVandEq to the current goal.
Assume HV: V ∈ Tx.
Assume HUeq: U = V ∩ Y.
We prove the intermediate claim HVY: V ∩ Y ∈ Tx.
An exact proof term for the current goal is (topology_binintersect_closed X Tx V Y HTx HV HY).
We prove the intermediate claim HUinTx: U ∈ Tx.
rewrite the current goal using HUeq (from left to right).
An exact proof term for the current goal is HVY.
An exact proof term for the current goal is HUinTx.
∎
Theorem. (product_subspace_topology)
∀X Tx Y Ty A B : set, topology_on X Tx → topology_on Y Ty → A ⊆ X → B ⊆ Y → product_topology A (subspace_topology X Tx A) B (subspace_topology Y Ty B) = subspace_topology (OrderedPair X Y) (product_topology X Tx Y Ty) (OrderedPair A B)
In Proofgold the corresponding term root is 3594ca... and proposition id is a9da86...
Proof:
Let X, Tx, Y, Ty, A and B be given.
Assume HTx: topology_on X Tx.
Assume HTy: topology_on Y Ty.
Assume HA: A ⊆ X.
Assume HB: B ⊆ Y.
We will prove product_topology A (subspace_topology X Tx A) B (subspace_topology Y Ty B) = subspace_topology (OrderedPair X Y) (product_topology X Tx Y Ty) (OrderedPair A B).
The rest of this subproof is missing.
∎
Definition. We define unit_interval to be R of type set.
In Proofgold the corresponding term root is e26ffa... and object id is 5277ef...
Definition. We define ordered_square to be OrderedPair unit_interval unit_interval of type set.
In Proofgold the corresponding term root is 60f7e1... and object id is d6fb09...
Definition. We define ordered_square_topology to be order_topology ordered_square of type set.
In Proofgold the corresponding term root is 9ce474... and object id is eb7cbd...
Definition. We define ordered_square_open_strip to be ordered_square of type set.
In Proofgold the corresponding term root is 60f7e1... and object id is d6fb09...
Definition. We define ordered_square_subspace_topology to be subspace_topology (OrderedPair R R) R2_dictionary_order_topology ordered_square of type set.
In Proofgold the corresponding term root is ad39ff... and object id is c3030f...
Theorem. (ordered_square_not_subspace_dictionary)
ordered_square_topology ≠ subspace_topology (OrderedPair R R) R2_dictionary_order_topology ordered_square
In Proofgold the corresponding term root is 3f658c... and proposition id is 1ff9f7...
Proof:
We will prove ordered_square_topology ≠ subspace_topology (OrderedPair R R) R2_dictionary_order_topology ordered_square.
The rest of this subproof is missing.
∎
Theorem. (convex_subspace_order_topology)
∀X Y : set, order_topology Y = subspace_topology X (order_topology X) Y
In Proofgold the corresponding term root is 49f07e... and proposition id is d57aac...
Proof:
Let X and Y be given.
We will prove order_topology Y = subspace_topology X (order_topology X) Y.
The rest of this subproof is missing.
∎
Theorem. (binintersect_right_absorb_subset)
∀W Y A : set, A ⊆ Y → (W ∩ Y) ∩ A = W ∩ A
In Proofgold the corresponding term root is 0d33fc... and proposition id is afcf75...
Proof:
Let W, Y and A be given.
Assume Hsub: A ⊆ Y.
Apply set_ext to the current goal.
Let x be given.
Assume Hx: x ∈ (W ∩ Y) ∩ A.
We prove the intermediate claim Hpair: x ∈ W ∩ Y ∧ x ∈ A.
An exact proof term for the current goal is (binintersectE (W ∩ Y) A x Hx).
We prove the intermediate claim HWY: x ∈ W ∩ Y.
An exact proof term for the current goal is (andEL (x ∈ W ∩ Y) (x ∈ A) Hpair).
We prove the intermediate claim HA: x ∈ A.
An exact proof term for the current goal is (andER (x ∈ W ∩ Y) (x ∈ A) Hpair).
We prove the intermediate claim HWYpair: x ∈ W ∧ x ∈ Y.
An exact proof term for the current goal is (binintersectE W Y x HWY).
We prove the intermediate claim HW: x ∈ W.
An exact proof term for the current goal is (andEL (x ∈ W) (x ∈ Y) HWYpair).
Apply binintersectI to the current goal.
An exact proof term for the current goal is HW.
An exact proof term for the current goal is HA.
Let x be given.
Assume Hx: x ∈ W ∩ A.
We prove the intermediate claim Hpair: x ∈ W ∧ x ∈ A.
An exact proof term for the current goal is (binintersectE W A x Hx).
We prove the intermediate claim HW: x ∈ W.
An exact proof term for the current goal is (andEL (x ∈ W) (x ∈ A) Hpair).
We prove the intermediate claim HA: x ∈ A.
An exact proof term for the current goal is (andER (x ∈ W) (x ∈ A) Hpair).
We prove the intermediate claim HY: x ∈ Y.
An exact proof term for the current goal is (Hsub x HA).
We prove the intermediate claim HWY: x ∈ W ∩ Y.
An exact proof term for the current goal is (binintersectI W Y x HW HY).
Apply binintersectI to the current goal.
An exact proof term for the current goal is HWY.
An exact proof term for the current goal is HA.
∎
Theorem. (ex16_1_subspace_transitive)
∀X Tx Y A : set, topology_on X Tx → Y ⊆ X → A ⊆ Y → subspace_topology Y (subspace_topology X Tx Y) A = subspace_topology X Tx A
In Proofgold the corresponding term root is 8ebb42... and proposition id is de9f7c...
Proof:
Let X, Tx, Y and A be given.
Assume Htop: topology_on X Tx.
Assume HY: Y ⊆ X.
Assume HA: A ⊆ Y.
We will prove subspace_topology Y (subspace_topology X Tx Y) A = subspace_topology X Tx A.
Apply set_ext to the current goal.
Let W be given.
Assume HW: W ∈ subspace_topology Y (subspace_topology X Tx Y) A.
We will prove W ∈ subspace_topology X Tx A.
We prove the intermediate claim HWPowerA: W ∈ 𝒫 A.
An exact proof term for the current goal is (SepE1 (𝒫 A) (λU0 : set ⇒ ∃U ∈ subspace_topology X Tx Y, U0 = U ∩ A) W HW).
We prove the intermediate claim HWexists: ∃U ∈ subspace_topology X Tx Y, W = U ∩ A.
An exact proof term for the current goal is (SepE2 (𝒫 A) (λU0 : set ⇒ ∃U ∈ subspace_topology X Tx Y, U0 = U ∩ A) W HW).
Apply HWexists to the current goal.
Let U be given.
Assume HU: U ∈ subspace_topology X Tx Y ∧ W = U ∩ A.
We prove the intermediate claim HUinSubY: U ∈ subspace_topology X Tx Y.
An exact proof term for the current goal is (andEL (U ∈ subspace_topology X Tx Y) (W = U ∩ A) HU).
We prove the intermediate claim HWeqUA: W = U ∩ A.
An exact proof term for the current goal is (andER (U ∈ subspace_topology X Tx Y) (W = U ∩ A) HU).
We prove the intermediate claim HUPowerY: U ∈ 𝒫 Y.
An exact proof term for the current goal is (SepE1 (𝒫 Y) (λU0 : set ⇒ ∃V ∈ Tx, U0 = V ∩ Y) U HUinSubY).
We prove the intermediate claim HUexists: ∃V ∈ Tx, U = V ∩ Y.
An exact proof term for the current goal is (SepE2 (𝒫 Y) (λU0 : set ⇒ ∃V ∈ Tx, U0 = V ∩ Y) U HUinSubY).
Apply HUexists to the current goal.
Let V be given.
Assume HV: V ∈ Tx ∧ U = V ∩ Y.
We prove the intermediate claim HVinTx: V ∈ Tx.
An exact proof term for the current goal is (andEL (V ∈ Tx) (U = V ∩ Y) HV).
We prove the intermediate claim HUeqVY: U = V ∩ Y.
An exact proof term for the current goal is (andER (V ∈ Tx) (U = V ∩ Y) HV).
We prove the intermediate claim HWeqVA: W = V ∩ A.
rewrite the current goal using HWeqUA (from left to right).
rewrite the current goal using HUeqVY (from left to right).
An exact proof term for the current goal is (binintersect_right_absorb_subset V Y A HA).
We prove the intermediate claim HWPred: ∃V0 ∈ Tx, W = V0 ∩ A.
We use V to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is HVinTx.
An exact proof term for the current goal is HWeqVA.
An exact proof term for the current goal is (SepI (𝒫 A) (λW0 : set ⇒ ∃V0 ∈ Tx, W0 = V0 ∩ A) W HWPowerA HWPred).
Let W be given.
Assume HW: W ∈ subspace_topology X Tx A.
We will prove W ∈ subspace_topology Y (subspace_topology X Tx Y) A.
We prove the intermediate claim HWPowerA: W ∈ 𝒫 A.
An exact proof term for the current goal is (SepE1 (𝒫 A) (λW0 : set ⇒ ∃V ∈ Tx, W0 = V ∩ A) W HW).
We prove the intermediate claim HWexists: ∃V ∈ Tx, W = V ∩ A.
An exact proof term for the current goal is (SepE2 (𝒫 A) (λW0 : set ⇒ ∃V ∈ Tx, W0 = V ∩ A) W HW).
Apply HWexists to the current goal.
Let V be given.
Assume HV: V ∈ Tx ∧ W = V ∩ A.
We prove the intermediate claim HVinTx: V ∈ Tx.
An exact proof term for the current goal is (andEL (V ∈ Tx) (W = V ∩ A) HV).
We prove the intermediate claim HWeqVA: W = V ∩ A.
An exact proof term for the current goal is (andER (V ∈ Tx) (W = V ∩ A) HV).
Set U to be the term V ∩ Y.
We prove the intermediate claim HUinSubY: U ∈ subspace_topology X Tx Y.
We prove the intermediate claim HUPowerY: U ∈ 𝒫 Y.
An exact proof term for the current goal is (PowerI Y U (binintersect_Subq_2 V Y)).
We prove the intermediate claim HUPred: ∃V0 ∈ Tx, U = V0 ∩ Y.
We use V to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is HVinTx.
Use reflexivity.
An exact proof term for the current goal is (SepI (𝒫 Y) (λU0 : set ⇒ ∃V0 ∈ Tx, U0 = V0 ∩ Y) U HUPowerY HUPred).
We prove the intermediate claim HWeqUA: W = U ∩ A.
rewrite the current goal using HWeqVA (from left to right).
Use symmetry.
An exact proof term for the current goal is (binintersect_right_absorb_subset V Y A HA).
We prove the intermediate claim HWPred: ∃U0 ∈ subspace_topology X Tx Y, W = U0 ∩ A.
We use U to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is HUinSubY.
An exact proof term for the current goal is HWeqUA.
An exact proof term for the current goal is (SepI (𝒫 A) (λW0 : set ⇒ ∃U0 ∈ subspace_topology X Tx Y, W0 = U0 ∩ A) W HWPowerA HWPred).
∎
Theorem. (ex16_2_finer_subspaces)
∀X T T' Y : set, topology_on X T → topology_on X T' → T' ⊆ T → Y ⊆ X → subspace_topology X T' Y ⊆ subspace_topology X T Y
In Proofgold the corresponding term root is 4e66a7... and proposition id is 74d02b...
Proof:
Let X, T, T' and Y be given.
Assume Htop: topology_on X T.
Assume Htop': topology_on X T'.
Assume Hfiner: T' ⊆ T.
Assume HY: Y ⊆ X.
We will prove subspace_topology X T' Y ⊆ subspace_topology X T Y.
Let W be given.
Assume HW: W ∈ subspace_topology X T' Y.
We will prove W ∈ subspace_topology X T Y.
We prove the intermediate claim HWPowerY: W ∈ 𝒫 Y.
An exact proof term for the current goal is (SepE1 (𝒫 Y) (λW0 : set ⇒ ∃V ∈ T', W0 = V ∩ Y) W HW).
We prove the intermediate claim HWexists: ∃V ∈ T', W = V ∩ Y.
An exact proof term for the current goal is (SepE2 (𝒫 Y) (λW0 : set ⇒ ∃V ∈ T', W0 = V ∩ Y) W HW).
Apply HWexists to the current goal.
Let V' be given.
Assume HV': V' ∈ T' ∧ W = V' ∩ Y.
We prove the intermediate claim HV'inT': V' ∈ T'.
An exact proof term for the current goal is (andEL (V' ∈ T') (W = V' ∩ Y) HV').
We prove the intermediate claim HWeqV'Y: W = V' ∩ Y.
An exact proof term for the current goal is (andER (V' ∈ T') (W = V' ∩ Y) HV').
We prove the intermediate claim HV'inT: V' ∈ T.
An exact proof term for the current goal is (Hfiner V' HV'inT').
We prove the intermediate claim HWPred: ∃V ∈ T, W = V ∩ Y.
We use V' to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is HV'inT.
An exact proof term for the current goal is HWeqV'Y.
An exact proof term for the current goal is (SepI (𝒫 Y) (λW0 : set ⇒ ∃V ∈ T, W0 = V ∩ Y) W HWPowerY HWPred).
∎
Definition. We define interval_A to be open_interval Empty (𝒫 Empty) of type set.
In Proofgold the corresponding term root is 016e3b... and object id is da5179...
Definition. We define interval_B to be open_interval (𝒫 Empty) (𝒫 (𝒫 Empty)) of type set.
In Proofgold the corresponding term root is e044f0... and object id is 37325c...
Definition. We define interval_C to be open_interval Empty Empty of type set.
In Proofgold the corresponding term root is 7f5a88... and object id is 295a42...
Definition. We define interval_D to be open_interval (𝒫 Empty) (𝒫 Empty) of type set.
In Proofgold the corresponding term root is b22711... and object id is 76f051...
Definition. We define interval_E to be open_interval (𝒫 (𝒫 Empty)) (𝒫 (𝒫 Empty)) of type set.
In Proofgold the corresponding term root is 5665e5... and object id is 5eb667...
Theorem. (ex16_3_open_sets_subspace)
∀X Tx Y : set, topology_on X Tx → Y ⊆ X → ∀U : set, open_in Y (subspace_topology X Tx Y) U → ∃V : set, open_in X Tx V ∧ U = V ∩ Y
In Proofgold the corresponding term root is b6764f... and proposition id is 7d86a7...
Proof:
Let X, Tx and Y be given.
Assume Htop: topology_on X Tx.
Assume HY: Y ⊆ X.
Let U be given.
Assume HU: open_in Y (subspace_topology X Tx Y) U.
We will prove ∃V : set, open_in X Tx V ∧ U = V ∩ Y.
We prove the intermediate claim HtopY: topology_on Y (subspace_topology X Tx Y).
An exact proof term for the current goal is (andEL (topology_on Y (subspace_topology X Tx Y)) (U ∈ subspace_topology X Tx Y) HU).
We prove the intermediate claim HUinSub: U ∈ subspace_topology X Tx Y.
An exact proof term for the current goal is (andER (topology_on Y (subspace_topology X Tx Y)) (U ∈ subspace_topology X Tx Y) HU).
We prove the intermediate claim HUPowerY: U ∈ 𝒫 Y.
An exact proof term for the current goal is (SepE1 (𝒫 Y) (λU0 : set ⇒ ∃V ∈ Tx, U0 = V ∩ Y) U HUinSub).
We prove the intermediate claim HUexists: ∃V ∈ Tx, U = V ∩ Y.
An exact proof term for the current goal is (SepE2 (𝒫 Y) (λU0 : set ⇒ ∃V ∈ Tx, U0 = V ∩ Y) U HUinSub).
Apply HUexists to the current goal.
Let V be given.
Assume HV: V ∈ Tx ∧ U = V ∩ Y.
We prove the intermediate claim HVinTx: V ∈ Tx.
An exact proof term for the current goal is (andEL (V ∈ Tx) (U = V ∩ Y) HV).
We prove the intermediate claim HUeqVY: U = V ∩ Y.
An exact proof term for the current goal is (andER (V ∈ Tx) (U = V ∩ Y) HV).
We prove the intermediate claim HVopen: open_in X Tx V.
An exact proof term for the current goal is (andI (topology_on X Tx) (V ∈ Tx) Htop HVinTx).
We use V to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is HVopen.
An exact proof term for the current goal is HUeqVY.
∎
Definition. We define projection_image1 to be λX Y U ⇒ {x ∈ X|∃y : set, OrderedPair x y ∈ U} of type set → set → set → set.
In Proofgold the corresponding term root is 7ea094... and object id is b76ba0...
Definition. We define projection_image2 to be λX Y U ⇒ {y ∈ Y|∃x : set, OrderedPair x y ∈ U} of type set → set → set → set.
In Proofgold the corresponding term root is 040f9d... and object id is ff3ea4...
Theorem. (ex16_4_projections_open)
∀X Tx Y Ty : set, topology_on X Tx → topology_on Y Ty → ∀U : set, U ∈ product_topology X Tx Y Ty → open_in X Tx (projection_image1 X Y U) ∧ open_in Y Ty (projection_image2 X Y U)
In Proofgold the corresponding term root is a4e834... and proposition id is 6b58e8...
Proof:
Let X, Tx, Y and Ty be given.
Assume HTx: topology_on X Tx.
Assume HTy: topology_on Y Ty.
Let U be given.
Assume HU: U ∈ product_topology X Tx Y Ty.
We will prove open_in X Tx (projection_image1 X Y U) ∧ open_in Y Ty (projection_image2 X Y U).
The rest of this subproof is missing.
∎
Theorem. (ex16_5a_product_monotone)
∀X T T' Y U U' : set, topology_on X T → topology_on X T' → topology_on Y U → topology_on Y U' → T ⊆ T' ∧ U ⊆ U' → product_topology X T Y U ⊆ product_topology X T' Y U'
In Proofgold the corresponding term root is d85937... and proposition id is aaa1d5...
Proof:
Let X, T, T', Y, U and U' be given.
Assume HTx: topology_on X T.
Assume HTx': topology_on X T'.
Assume HTy: topology_on Y U.
Assume HTy': topology_on Y U'.
Assume Hfiner: T ⊆ T' ∧ U ⊆ U'.
We will prove product_topology X T Y U ⊆ product_topology X T' Y U'.
The rest of this subproof is missing.
∎
Theorem. (ex16_5b_product_converse)
∀X T T' Y U U' : set, topology_on X T → topology_on X T' → topology_on Y U → topology_on Y U' → product_topology X T Y U ⊆ product_topology X T' Y U' → T ⊆ T' ∧ U ⊆ U'
In Proofgold the corresponding term root is 44a39a... and proposition id is 03e6f6...
Proof:
Let X, T, T', Y, U and U' be given.
Assume HTx: topology_on X T.
Assume HTx': topology_on X T'.
Assume HTy: topology_on Y U.
Assume HTy': topology_on Y U'.
Assume Hprod: product_topology X T Y U ⊆ product_topology X T' Y U'.
We will prove T ⊆ T' ∧ U ⊆ U'.
The rest of this subproof is missing.
∎
Definition. We define rational_rectangle_basis to be {r ∈ 𝒫 (OrderedPair R R)|∃a b c d : set, a ∈ rational_numbers ∧ b ∈ rational_numbers ∧ c ∈ rational_numbers ∧ d ∈ rational_numbers ∧ r = OrderedPair (open_interval a b) (open_interval c d)} of type set.
In Proofgold the corresponding term root is 42c555... and object id is 20a0d1...
Theorem. (ex16_6_rational_rectangles_basis)
basis_on (OrderedPair R R) rational_rectangle_basis ∧ generated_topology (OrderedPair R R) rational_rectangle_basis = R2_standard_topology
In Proofgold the corresponding term root is e77106... and proposition id is 899e73...
Definition. We define convex_subset to be λA ⇒ A ⊆ R ∧ ∀x y : set, x ∈ A → y ∈ A → open_interval x y ⊆ A of type set → prop.
In Proofgold the corresponding term root is 08850c... and object id is 4eb769...
Theorem. (ex16_7_convex_interval_or_ray)
∀A : set, convex_subset A → (A = Empty ∨ A = R ∨ ∃a b : set, A = open_interval a b ∨ A = {x ∈ R|Rlt a x} ∨ A = {x ∈ R|Rlt x b})
In Proofgold the corresponding term root is 7de140... and proposition id is f3c95f...
Proof:
Let A be given.
Assume HA: convex_subset A.
We will prove A = Empty ∨ A = R ∨ ∃a b : set, A = open_interval a b ∨ A = {x ∈ R|Rlt a x} ∨ A = {x ∈ R|Rlt x b}.
The rest of this subproof is missing.
∎
Theorem. (ex16_8_lines_in_lower_limit_products)
∃L : set, L ⊆ OrderedPair R R ∧ L = {OrderedPair x x|x ∈ R} ∧ subspace_topology (OrderedPair R R) (product_topology R R_lower_limit_topology R R_lower_limit_topology) L = R_lower_limit_topology
In Proofgold the corresponding term root is a9b6c2... and proposition id is 08db2d...
Theorem. (ex16_9_dictionary_equals_product)
R2_dictionary_order_topology ≠ product_topology R R_standard_topology R R_standard_topology
In Proofgold the corresponding term root is 95211e... and proposition id is 7dc389...
Proof:
We will prove R2_dictionary_order_topology ≠ product_topology R R_standard_topology R R_standard_topology.
The rest of this subproof is missing.
∎
Theorem. (ex16_10_compare_topologies_on_square)
ordered_square_topology ≠ subspace_topology (OrderedPair R R) R2_dictionary_order_topology ordered_square ∧ subspace_topology (OrderedPair R R) R2_dictionary_order_topology ordered_square ≠ product_topology unit_interval R_standard_topology unit_interval R_standard_topology
In Proofgold the corresponding term root is 994a8f... and proposition id is b45676...
Definition. We define interior_of to be λX T A ⇒ {x ∈ X|∃U : set, U ∈ T ∧ x ∈ U ∧ U ⊆ A} of type set → set → set → set.
In Proofgold the corresponding term root is 3b8bd0... and object id is 0b3836...
Definition. We define closure_of to be λX T A ⇒ {x ∈ X|∀U : set, U ∈ T → x ∈ U → U ∩ A ≠ Empty} of type set → set → set → set.
In Proofgold the corresponding term root is 66761a... and object id is 355f66...
Theorem. (subset_of_closure)
∀X Tx A : set, topology_on X Tx → A ⊆ X → A ⊆ closure_of X Tx A
In Proofgold the corresponding term root is 8bb62a... and proposition id is 34e2cf...
Proof:
Let X, Tx and A be given.
Assume Htop: topology_on X Tx.
Assume HA: A ⊆ X.
We will prove A ⊆ closure_of X Tx A.
Let x be given.
Assume Hx: x ∈ A.
We will prove x ∈ closure_of X Tx A.
We prove the intermediate claim HxX: x ∈ X.
An exact proof term for the current goal is (HA x Hx).
We prove the intermediate claim Hcond: ∀U : set, U ∈ Tx → x ∈ U → U ∩ A ≠ Empty.
Let U be given.
Assume HU: U ∈ Tx.
Assume HxU: x ∈ U.
We will prove U ∩ A ≠ Empty.
Assume Hempty: U ∩ A = Empty.
We prove the intermediate claim HxUA: x ∈ U ∩ A.
An exact proof term for the current goal is (binintersectI U A x HxU Hx).
We prove the intermediate claim HxEmpty: x ∈ Empty.
rewrite the current goal using Hempty (from right to left).
An exact proof term for the current goal is HxUA.
An exact proof term for the current goal is (EmptyE x HxEmpty).
An exact proof term for the current goal is (SepI X (λx0 ⇒ ∀U : set, U ∈ Tx → x0 ∈ U → U ∩ A ≠ Empty) x HxX Hcond).
∎
Theorem. (closure_monotone)
∀X Tx A B : set, topology_on X Tx → A ⊆ B → B ⊆ X → closure_of X Tx A ⊆ closure_of X Tx B
In Proofgold the corresponding term root is 7e4362... and proposition id is 96a594...
Proof:
Let X, Tx, A and B be given.
Assume Htop: topology_on X Tx.
Assume HAB: A ⊆ B.
Assume HB: B ⊆ X.
We will prove closure_of X Tx A ⊆ closure_of X Tx B.
Let x be given.
Assume Hx: x ∈ closure_of X Tx A.
We will prove x ∈ closure_of X Tx B.
We prove the intermediate claim HxX: x ∈ X.
An exact proof term for the current goal is (SepE1 X (λx0 ⇒ ∀U : set, U ∈ Tx → x0 ∈ U → U ∩ A ≠ Empty) x Hx).
We prove the intermediate claim HcondA: ∀U : set, U ∈ Tx → x ∈ U → U ∩ A ≠ Empty.
An exact proof term for the current goal is (SepE2 X (λx0 ⇒ ∀U : set, U ∈ Tx → x0 ∈ U → U ∩ A ≠ Empty) x Hx).
We prove the intermediate claim HcondB: ∀U : set, U ∈ Tx → x ∈ U → U ∩ B ≠ Empty.
Let U be given.
Assume HU: U ∈ Tx.
Assume HxU: x ∈ U.
We will prove U ∩ B ≠ Empty.
We prove the intermediate claim HUA_ne: U ∩ A ≠ Empty.
An exact proof term for the current goal is (HcondA U HU HxU).
Assume Hempty: U ∩ B = Empty.
We prove the intermediate claim HUA_sub_UB: U ∩ A ⊆ U ∩ B.
Let y be given.
Assume Hy: y ∈ U ∩ A.
We prove the intermediate claim HyU: y ∈ U.
An exact proof term for the current goal is (binintersectE1 U A y Hy).
We prove the intermediate claim HyA: y ∈ A.
An exact proof term for the current goal is (binintersectE2 U A y Hy).
We prove the intermediate claim HyB: y ∈ B.
An exact proof term for the current goal is (HAB y HyA).
An exact proof term for the current goal is (binintersectI U B y HyU HyB).
We prove the intermediate claim HUA_empty: U ∩ A = Empty.
Apply Empty_Subq_eq to the current goal.
We prove the intermediate claim HUB_sub_Empty: U ∩ B ⊆ Empty.
rewrite the current goal using Hempty (from left to right).
An exact proof term for the current goal is (Subq_ref Empty).
An exact proof term for the current goal is (Subq_tra (U ∩ A) (U ∩ B) Empty HUA_sub_UB HUB_sub_Empty).
An exact proof term for the current goal is (HUA_ne HUA_empty).
An exact proof term for the current goal is (SepI X (λx0 ⇒ ∀U : set, U ∈ Tx → x0 ∈ U → U ∩ B ≠ Empty) x HxX HcondB).
∎
Theorem. (interior_subset)
∀X Tx A : set, topology_on X Tx → interior_of X Tx A ⊆ A
In Proofgold the corresponding term root is d5f872... and proposition id is 2bc9c5...
Proof:
Let X, Tx and A be given.
Assume Htop: topology_on X Tx.
We will prove interior_of X Tx A ⊆ A.
Let x be given.
Assume Hx: x ∈ interior_of X Tx A.
We will prove x ∈ A.
We prove the intermediate claim Hexists: ∃U : set, U ∈ Tx ∧ x ∈ U ∧ U ⊆ A.
An exact proof term for the current goal is (SepE2 X (λx0 ⇒ ∃U : set, U ∈ Tx ∧ x0 ∈ U ∧ U ⊆ A) x Hx).
Apply Hexists to the current goal.
Let U be given.
Assume HU_conj: U ∈ Tx ∧ x ∈ U ∧ U ⊆ A.
We prove the intermediate claim HU_and_x: U ∈ Tx ∧ x ∈ U.
An exact proof term for the current goal is (andEL (U ∈ Tx ∧ x ∈ U) (U ⊆ A) HU_conj).
We prove the intermediate claim HUsub: U ⊆ A.
An exact proof term for the current goal is (andER (U ∈ Tx ∧ x ∈ U) (U ⊆ A) HU_conj).
We prove the intermediate claim HxU: x ∈ U.
An exact proof term for the current goal is (andER (U ∈ Tx) (x ∈ U) HU_and_x).
An exact proof term for the current goal is (HUsub x HxU).
∎
Theorem. (interior_monotone)
∀X Tx A B : set, topology_on X Tx → A ⊆ B → interior_of X Tx A ⊆ interior_of X Tx B
In Proofgold the corresponding term root is 9e824d... and proposition id is 7aee49...
Proof:
Let X, Tx, A and B be given.
Assume Htop: topology_on X Tx.
Assume HAB: A ⊆ B.
We will prove interior_of X Tx A ⊆ interior_of X Tx B.
Let x be given.
Assume Hx: x ∈ interior_of X Tx A.
We will prove x ∈ interior_of X Tx B.
We prove the intermediate claim HxX: x ∈ X.
An exact proof term for the current goal is (SepE1 X (λx0 ⇒ ∃U : set, U ∈ Tx ∧ x0 ∈ U ∧ U ⊆ A) x Hx).
We prove the intermediate claim Hexists: ∃U : set, U ∈ Tx ∧ x ∈ U ∧ U ⊆ A.
An exact proof term for the current goal is (SepE2 X (λx0 ⇒ ∃U : set, U ∈ Tx ∧ x0 ∈ U ∧ U ⊆ A) x Hx).
Apply Hexists to the current goal.
Let U be given.
Assume HU_conj: U ∈ Tx ∧ x ∈ U ∧ U ⊆ A.
We prove the intermediate claim HU_and_x: U ∈ Tx ∧ x ∈ U.
An exact proof term for the current goal is (andEL (U ∈ Tx ∧ x ∈ U) (U ⊆ A) HU_conj).
We prove the intermediate claim HUsub_A: U ⊆ A.
An exact proof term for the current goal is (andER (U ∈ Tx ∧ x ∈ U) (U ⊆ A) HU_conj).
We prove the intermediate claim HU: U ∈ Tx.
An exact proof term for the current goal is (andEL (U ∈ Tx) (x ∈ U) HU_and_x).
We prove the intermediate claim HxU: x ∈ U.
An exact proof term for the current goal is (andER (U ∈ Tx) (x ∈ U) HU_and_x).
We prove the intermediate claim HUsub_B: U ⊆ B.
An exact proof term for the current goal is (Subq_tra U A B HUsub_A HAB).
We prove the intermediate claim Hwitness: U ∈ Tx ∧ x ∈ U ∧ U ⊆ B.
Apply andI to the current goal.
Apply andI to the current goal.
An exact proof term for the current goal is HU.
An exact proof term for the current goal is HxU.
An exact proof term for the current goal is HUsub_B.
We prove the intermediate claim Hexists_B: ∃V : set, V ∈ Tx ∧ x ∈ V ∧ V ⊆ B.
We use U to witness the existential quantifier.
An exact proof term for the current goal is Hwitness.
An exact proof term for the current goal is (SepI X (λx0 ⇒ ∃V : set, V ∈ Tx ∧ x0 ∈ V ∧ V ⊆ B) x HxX Hexists_B).
∎
Theorem. (open_interior_eq)
∀X Tx U : set, topology_on X Tx → U ∈ Tx → interior_of X Tx U = U
In Proofgold the corresponding term root is 2231ac... and proposition id is 9e1c33...
Proof:
Let X, Tx and U be given.
Assume Htop: topology_on X Tx.
Assume HU: U ∈ Tx.
We will prove interior_of X Tx U = U.
Apply set_ext to the current goal.
An exact proof term for the current goal is (interior_subset X Tx U Htop).
Let x be given.
Assume Hx: x ∈ U.
We will prove x ∈ interior_of X Tx U.
We prove the intermediate claim HUsub: U ⊆ X.
An exact proof term for the current goal is (topology_elem_subset X Tx U Htop HU).
We prove the intermediate claim HxX: x ∈ X.
An exact proof term for the current goal is (HUsub x Hx).
We prove the intermediate claim Hwitness: U ∈ Tx ∧ x ∈ U ∧ U ⊆ U.
Apply andI to the current goal.
Apply andI to the current goal.
An exact proof term for the current goal is HU.
An exact proof term for the current goal is Hx.
An exact proof term for the current goal is (Subq_ref U).
We prove the intermediate claim Hexists: ∃V : set, V ∈ Tx ∧ x ∈ V ∧ V ⊆ U.
We use U to witness the existential quantifier.
An exact proof term for the current goal is Hwitness.
An exact proof term for the current goal is (SepI X (λx0 ⇒ ∃V : set, V ∈ Tx ∧ x0 ∈ V ∧ V ⊆ U) x HxX Hexists).
∎
Theorem. (interior_of_empty)
∀X Tx : set, topology_on X Tx → interior_of X Tx Empty = Empty
In Proofgold the corresponding term root is 744616... and proposition id is ca8b2d...
Proof:
Let X and Tx be given.
Assume Htop: topology_on X Tx.
We will prove interior_of X Tx Empty = Empty.
Apply set_ext to the current goal.
An exact proof term for the current goal is (interior_subset X Tx Empty Htop).
An exact proof term for the current goal is (Subq_Empty (interior_of X Tx Empty)).
∎
Theorem. (interior_of_space)
∀X Tx : set, topology_on X Tx → interior_of X Tx X = X
In Proofgold the corresponding term root is f0c83b... and proposition id is bfa24d...
Proof:
Let X and Tx be given.
Assume Htop: topology_on X Tx.
We will prove interior_of X Tx X = X.
We prove the intermediate claim HXopen: X ∈ Tx.
An exact proof term for the current goal is (topology_has_X X Tx Htop).
An exact proof term for the current goal is (open_interior_eq X Tx X Htop HXopen).
∎
Theorem. (interior_is_open)
∀X Tx A : set, topology_on X Tx → A ⊆ X → interior_of X Tx A ∈ Tx
In Proofgold the corresponding term root is 1e05a3... and proposition id is 420da6...
Proof:
Let X, Tx and A be given.
Assume Htop: topology_on X Tx.
Assume HA: A ⊆ X.
We will prove interior_of X Tx A ∈ Tx.
Set F to be the term {U ∈ Tx|U ⊆ A}.
We prove the intermediate claim Hint_eq_union: interior_of X Tx A = ⋃ F.
Apply set_ext to the current goal.
Let x be given.
Assume Hx: x ∈ interior_of X Tx A.
We will prove x ∈ ⋃ F.
We prove the intermediate claim HxX: x ∈ X.
An exact proof term for the current goal is (SepE1 X (λx0 ⇒ ∃U : set, U ∈ Tx ∧ x0 ∈ U ∧ U ⊆ A) x Hx).
We prove the intermediate claim Hexists: ∃U : set, U ∈ Tx ∧ x ∈ U ∧ U ⊆ A.
An exact proof term for the current goal is (SepE2 X (λx0 ⇒ ∃U : set, U ∈ Tx ∧ x0 ∈ U ∧ U ⊆ A) x Hx).
Apply Hexists to the current goal.
Let U be given.
Assume HU_conj: U ∈ Tx ∧ x ∈ U ∧ U ⊆ A.
We prove the intermediate claim HU_and_x: U ∈ Tx ∧ x ∈ U.
An exact proof term for the current goal is (andEL (U ∈ Tx ∧ x ∈ U) (U ⊆ A) HU_conj).
We prove the intermediate claim HUsub: U ⊆ A.
An exact proof term for the current goal is (andER (U ∈ Tx ∧ x ∈ U) (U ⊆ A) HU_conj).
We prove the intermediate claim HU: U ∈ Tx.
An exact proof term for the current goal is (andEL (U ∈ Tx) (x ∈ U) HU_and_x).
We prove the intermediate claim HxU: x ∈ U.
An exact proof term for the current goal is (andER (U ∈ Tx) (x ∈ U) HU_and_x).
We prove the intermediate claim HUinF: U ∈ F.
Apply SepI to the current goal.
An exact proof term for the current goal is HU.
An exact proof term for the current goal is HUsub.
An exact proof term for the current goal is (UnionI F x U HxU HUinF).
Let x be given.
Assume Hx: x ∈ ⋃ F.
We will prove x ∈ interior_of X Tx A.
Apply (UnionE_impred F x Hx (x ∈ interior_of X Tx A)) to the current goal.
Let U be given.
Assume HxU: x ∈ U.
Assume HUinF: U ∈ F.
We prove the intermediate claim HU: U ∈ Tx.
An exact proof term for the current goal is (SepE1 Tx (λU0 ⇒ U0 ⊆ A) U HUinF).
We prove the intermediate claim HUsub: U ⊆ A.
An exact proof term for the current goal is (SepE2 Tx (λU0 ⇒ U0 ⊆ A) U HUinF).
We prove the intermediate claim HUsub_X: U ⊆ X.
An exact proof term for the current goal is (topology_elem_subset X Tx U Htop HU).
We prove the intermediate claim HxX: x ∈ X.
An exact proof term for the current goal is (HUsub_X x HxU).
We prove the intermediate claim Hwitness: U ∈ Tx ∧ x ∈ U ∧ U ⊆ A.
Apply andI to the current goal.
Apply andI to the current goal.
An exact proof term for the current goal is HU.
An exact proof term for the current goal is HxU.
An exact proof term for the current goal is HUsub.
We prove the intermediate claim Hexists: ∃V : set, V ∈ Tx ∧ x ∈ V ∧ V ⊆ A.
We use U to witness the existential quantifier.
An exact proof term for the current goal is Hwitness.
An exact proof term for the current goal is (SepI X (λx0 ⇒ ∃V : set, V ∈ Tx ∧ x0 ∈ V ∧ V ⊆ A) x HxX Hexists).
We prove the intermediate claim HF_sub_Tx: F ⊆ Tx.
Let U be given.
Assume HU: U ∈ F.
An exact proof term for the current goal is (SepE1 Tx (λU0 ⇒ U0 ⊆ A) U HU).
We prove the intermediate claim Hunion_in_Tx: ⋃ F ∈ Tx.
An exact proof term for the current goal is (topology_union_closed X Tx F Htop HF_sub_Tx).
The rest of this subproof is missing.
∎
Theorem. (interior_union_contains_union_interiors)
∀X Tx A B : set, topology_on X Tx → A ⊆ X → B ⊆ X → interior_of X Tx A ∪ interior_of X Tx B ⊆ interior_of X Tx (A ∪ B)
In Proofgold the corresponding term root is 2f3def... and proposition id is 873cfd...
Proof:
Let X, Tx, A and B be given.
Assume Htop: topology_on X Tx.
Assume HA: A ⊆ X.
Assume HB: B ⊆ X.
We will prove interior_of X Tx A ∪ interior_of X Tx B ⊆ interior_of X Tx (A ∪ B).
We prove the intermediate claim HAB_union_A: A ⊆ A ∪ B.
Let x be given.
Assume Hx: x ∈ A.
An exact proof term for the current goal is (binunionI1 A B x Hx).
We prove the intermediate claim HAB_union_B: B ⊆ A ∪ B.
Let x be given.
Assume Hx: x ∈ B.
An exact proof term for the current goal is (binunionI2 A B x Hx).
We prove the intermediate claim HAB_sub: A ∪ B ⊆ X.
Let x be given.
Assume Hx: x ∈ A ∪ B.
Apply (binunionE A B x Hx) to the current goal.
Assume HxA: x ∈ A.
An exact proof term for the current goal is (HA x HxA).
Assume HxB: x ∈ B.
An exact proof term for the current goal is (HB x HxB).
We prove the intermediate claim HintA: interior_of X Tx A ⊆ interior_of X Tx (A ∪ B).
An exact proof term for the current goal is (interior_monotone X Tx A (A ∪ B) Htop HAB_union_A).
We prove the intermediate claim HintB: interior_of X Tx B ⊆ interior_of X Tx (A ∪ B).
An exact proof term for the current goal is (interior_monotone X Tx B (A ∪ B) Htop HAB_union_B).
Let x be given.
Assume Hx: x ∈ interior_of X Tx A ∪ interior_of X Tx B.
Apply (binunionE (interior_of X Tx A) (interior_of X Tx B) x Hx) to the current goal.
Assume HxA: x ∈ interior_of X Tx A.
An exact proof term for the current goal is (HintA x HxA).
Assume HxB: x ∈ interior_of X Tx B.
An exact proof term for the current goal is (HintB x HxB).
∎
Theorem. (interior_intersection_contains_intersection)
∀X Tx A B : set, topology_on X Tx → A ⊆ X → B ⊆ X → interior_of X Tx (A ∩ B) ⊆ interior_of X Tx A ∩ interior_of X Tx B
In Proofgold the corresponding term root is 50a55e... and proposition id is e9f03f...
Proof:
Let X, Tx, A and B be given.
Assume Htop: topology_on X Tx.
Assume HA: A ⊆ X.
Assume HB: B ⊆ X.
We will prove interior_of X Tx (A ∩ B) ⊆ interior_of X Tx A ∩ interior_of X Tx B.
We prove the intermediate claim HAB_A: A ∩ B ⊆ A.
An exact proof term for the current goal is (binintersect_Subq_1 A B).
We prove the intermediate claim HAB_B: A ∩ B ⊆ B.
An exact proof term for the current goal is (binintersect_Subq_2 A B).
We prove the intermediate claim HintAB_A: interior_of X Tx (A ∩ B) ⊆ interior_of X Tx A.
An exact proof term for the current goal is (interior_monotone X Tx (A ∩ B) A Htop HAB_A).
We prove the intermediate claim HintAB_B: interior_of X Tx (A ∩ B) ⊆ interior_of X Tx B.
An exact proof term for the current goal is (interior_monotone X Tx (A ∩ B) B Htop HAB_B).
Let x be given.
Assume Hx: x ∈ interior_of X Tx (A ∩ B).
Apply binintersectI to the current goal.
An exact proof term for the current goal is (HintAB_A x Hx).
An exact proof term for the current goal is (HintAB_B x Hx).
∎
Theorem. (interior_idempotent)
∀X Tx A : set, topology_on X Tx → A ⊆ X → interior_of X Tx (interior_of X Tx A) = interior_of X Tx A
In Proofgold the corresponding term root is bf6e75... and proposition id is f33a2a...
Proof:
Let X, Tx and A be given.
Assume Htop: topology_on X Tx.
Assume HA: A ⊆ X.
We will prove interior_of X Tx (interior_of X Tx A) = interior_of X Tx A.
We prove the intermediate claim HintA_open: interior_of X Tx A ∈ Tx.
An exact proof term for the current goal is (interior_is_open X Tx A Htop HA).
An exact proof term for the current goal is (open_interior_eq X Tx (interior_of X Tx A) Htop HintA_open).
∎
Theorem. (closure_contains_set)
∀X Tx A : set, topology_on X Tx → A ⊆ X → A ⊆ closure_of X Tx A
In Proofgold the corresponding term root is 8bb62a... and proposition id is 34e2cf...
Proof:
Let X, Tx and A be given.
Assume Htop: topology_on X Tx.
Assume HA: A ⊆ X.
An exact proof term for the current goal is (subset_of_closure X Tx A Htop HA).
∎
Theorem. (closure_in_space)
∀X Tx A : set, topology_on X Tx → closure_of X Tx A ⊆ X
In Proofgold the corresponding term root is f74310... and proposition id is 9643ef...
Proof:
Let X, Tx and A be given.
Assume Htop: topology_on X Tx.
We will prove closure_of X Tx A ⊆ X.
Let x be given.
Assume Hx: x ∈ closure_of X Tx A.
We will prove x ∈ X.
An exact proof term for the current goal is (SepE1 X (λx0 ⇒ ∀U : set, U ∈ Tx → x0 ∈ U → U ∩ A ≠ Empty) x Hx).
∎
Theorem. (closure_union_contains_union_closures)
∀X Tx A B : set, topology_on X Tx → A ⊆ X → B ⊆ X → closure_of X Tx A ∪ closure_of X Tx B ⊆ closure_of X Tx (A ∪ B)
In Proofgold the corresponding term root is bc6216... and proposition id is a03032...
Proof:
Let X, Tx, A and B be given.
Assume Htop: topology_on X Tx.
Assume HA: A ⊆ X.
Assume HB: B ⊆ X.
We will prove closure_of X Tx A ∪ closure_of X Tx B ⊆ closure_of X Tx (A ∪ B).
We prove the intermediate claim HAB_union: A ⊆ A ∪ B.
Let x be given.
Assume Hx: x ∈ A.
An exact proof term for the current goal is (binunionI1 A B x Hx).
We prove the intermediate claim HBB_union: B ⊆ A ∪ B.
Let x be given.
Assume Hx: x ∈ B.
An exact proof term for the current goal is (binunionI2 A B x Hx).
We prove the intermediate claim HAB_sub: A ∪ B ⊆ X.
Let x be given.
Assume Hx: x ∈ A ∪ B.
Apply (binunionE A B x Hx) to the current goal.
Assume HxA: x ∈ A.
An exact proof term for the current goal is (HA x HxA).
Assume HxB: x ∈ B.
An exact proof term for the current goal is (HB x HxB).
We prove the intermediate claim HclA: closure_of X Tx A ⊆ closure_of X Tx (A ∪ B).
An exact proof term for the current goal is (closure_monotone X Tx A (A ∪ B) Htop HAB_union HAB_sub).
We prove the intermediate claim HclB: closure_of X Tx B ⊆ closure_of X Tx (A ∪ B).
An exact proof term for the current goal is (closure_monotone X Tx B (A ∪ B) Htop HBB_union HAB_sub).
Let x be given.
Assume Hx: x ∈ closure_of X Tx A ∪ closure_of X Tx B.
Apply (binunionE (closure_of X Tx A) (closure_of X Tx B) x Hx) to the current goal.
Assume HxA: x ∈ closure_of X Tx A.
An exact proof term for the current goal is (HclA x HxA).
Assume HxB: x ∈ closure_of X Tx B.
An exact proof term for the current goal is (HclB x HxB).
∎
Theorem. (closure_of_empty)
∀X Tx : set, topology_on X Tx → closure_of X Tx Empty = Empty
In Proofgold the corresponding term root is 0431e4... and proposition id is 064815...
Proof:
Let X and Tx be given.
Assume Htop: topology_on X Tx.
We will prove closure_of X Tx Empty = Empty.
Apply set_ext to the current goal.
We will prove closure_of X Tx Empty ⊆ Empty.
Let x be given.
Assume Hx: x ∈ closure_of X Tx Empty.
We will prove x ∈ Empty.
Apply (SepE X (λx0 ⇒ ∀U : set, U ∈ Tx → x0 ∈ U → U ∩ Empty ≠ Empty) x Hx) to the current goal.
Assume HxX: x ∈ X.
Assume Hcond: ∀U : set, U ∈ Tx → x ∈ U → U ∩ Empty ≠ Empty.
We prove the intermediate claim HXopen: X ∈ Tx.
An exact proof term for the current goal is (topology_has_X X Tx Htop).
We prove the intermediate claim HXne: X ∩ Empty ≠ Empty.
An exact proof term for the current goal is (Hcond X HXopen HxX).
We prove the intermediate claim HXempty: X ∩ Empty = Empty.
Apply set_ext to the current goal.
Let y be given.
Assume Hy: y ∈ X ∩ Empty.
An exact proof term for the current goal is (binintersectE2 X Empty y Hy).
An exact proof term for the current goal is (Subq_Empty (X ∩ Empty)).
Apply HXne to the current goal.
An exact proof term for the current goal is HXempty.
An exact proof term for the current goal is (Subq_Empty (closure_of X Tx Empty)).
∎
Theorem. (closure_of_space)
∀X Tx : set, topology_on X Tx → closure_of X Tx X = X
In Proofgold the corresponding term root is 34e1a3... and proposition id is 3f7d0c...
Proof:
Let X and Tx be given.
Assume Htop: topology_on X Tx.
We will prove closure_of X Tx X = X.
Apply set_ext to the current goal.
An exact proof term for the current goal is (closure_in_space X Tx X Htop).
An exact proof term for the current goal is (subset_of_closure X Tx X Htop (Subq_ref X)).
∎
Theorem. (closure_idempotent)
∀X Tx A : set, topology_on X Tx → A ⊆ X → closure_of X Tx (closure_of X Tx A) = closure_of X Tx A
In Proofgold the corresponding term root is 11169d... and proposition id is 68da76...
Proof:
Let X, Tx and A be given.
Assume Htop: topology_on X Tx.
Assume HA: A ⊆ X.
We will prove closure_of X Tx (closure_of X Tx A) = closure_of X Tx A.
We prove the intermediate claim HclA_sub: closure_of X Tx A ⊆ X.
An exact proof term for the current goal is (closure_in_space X Tx A Htop).
Apply set_ext to the current goal.
The rest of this subproof is missing.
An exact proof term for the current goal is (subset_of_closure X Tx (closure_of X Tx A) Htop HclA_sub).
∎
Theorem. (closure_intersection_contained)
∀X Tx A B : set, topology_on X Tx → A ⊆ X → B ⊆ X → closure_of X Tx (A ∩ B) ⊆ closure_of X Tx A ∩ closure_of X Tx B
In Proofgold the corresponding term root is 272d8b... and proposition id is 3106eb...
Proof:
Let X, Tx, A and B be given.
Assume Htop: topology_on X Tx.
Assume HA: A ⊆ X.
Assume HB: B ⊆ X.
We will prove closure_of X Tx (A ∩ B) ⊆ closure_of X Tx A ∩ closure_of X Tx B.
We prove the intermediate claim HAB_A: A ∩ B ⊆ A.
An exact proof term for the current goal is (binintersect_Subq_1 A B).
We prove the intermediate claim HAB_B: A ∩ B ⊆ B.
An exact proof term for the current goal is (binintersect_Subq_2 A B).
We prove the intermediate claim HAB_X: A ∩ B ⊆ X.
Let x be given.
Assume Hx: x ∈ A ∩ B.
An exact proof term for the current goal is (HA x (binintersectE1 A B x Hx)).
We prove the intermediate claim HclAB_A: closure_of X Tx (A ∩ B) ⊆ closure_of X Tx A.
An exact proof term for the current goal is (closure_monotone X Tx (A ∩ B) A Htop HAB_A HA).
We prove the intermediate claim HclAB_B: closure_of X Tx (A ∩ B) ⊆ closure_of X Tx B.
An exact proof term for the current goal is (closure_monotone X Tx (A ∩ B) B Htop HAB_B HB).
Let x be given.
Assume Hx: x ∈ closure_of X Tx (A ∩ B).
Apply binintersectI to the current goal.
An exact proof term for the current goal is (HclAB_A x Hx).
An exact proof term for the current goal is (HclAB_B x Hx).
∎
Theorem. (closed_closure_eq)
∀X Tx C : set, topology_on X Tx → closed_in X Tx C → closure_of X Tx C = C
In Proofgold the corresponding term root is 7a3894... and proposition id is 8ff2b2...
Proof:
Let X, Tx and C be given.
Assume Htop: topology_on X Tx.
Assume HC: closed_in X Tx C.
We will prove closure_of X Tx C = C.
We prove the intermediate claim HTx: topology_on X Tx.
An exact proof term for the current goal is (andEL (topology_on X Tx) (C ⊆ X ∧ ∃U ∈ Tx, C = X ∖ U) HC).
We prove the intermediate claim HCsub_and_ex: C ⊆ X ∧ ∃U ∈ Tx, C = X ∖ U.
An exact proof term for the current goal is (andER (topology_on X Tx) (C ⊆ X ∧ ∃U ∈ Tx, C = X ∖ U) HC).
We prove the intermediate claim HCsub: C ⊆ X.
An exact proof term for the current goal is (andEL (C ⊆ X) (∃U ∈ Tx, C = X ∖ U) HCsub_and_ex).
Apply set_ext to the current goal.
We will prove closure_of X Tx C ⊆ C.
The rest of this subproof is missing.
An exact proof term for the current goal is (subset_of_closure X Tx C Htop HCsub).
∎
Theorem. (closed_sets_axioms)
∀X T : set, topology_on X T → let C ≔ {X ∖ U|U ∈ T} in X ∈ C ∧ Empty ∈ C ∧ (∀F : set, F ∈ 𝒫 C → intersection_of_family F ∈ C) ∧ (∀A B : set, A ∈ C → B ∈ C → A ∪ B ∈ C)
In Proofgold the corresponding term root is b1e345... and proposition id is 91e35b...
Proof:
Let X and T be given.
Assume HT: topology_on X T.
We will prove let C ≔ {X ∖ U|U ∈ T} in X ∈ C ∧ Empty ∈ C ∧ (∀F : set, F ∈ 𝒫 C → intersection_of_family F ∈ C) ∧ (∀A B : set, A ∈ C → B ∈ C → A ∪ B ∈ C).
The rest of this subproof is missing.
∎
Theorem. (closed_in_subspace_iff_intersection)
∀X Tx Y A : set, topology_on X Tx → Y ⊆ X → (closed_in Y (subspace_topology X Tx Y) A ↔ ∃C : set, closed_in X Tx C ∧ A = C ∩ Y)
In Proofgold the corresponding term root is be4297... and proposition id is f144d1...
Proof:
Let X, Tx, Y and A be given.
Assume HTx: topology_on X Tx.
Assume HY: Y ⊆ X.
We will prove closed_in Y (subspace_topology X Tx Y) A ↔ ∃C : set, closed_in X Tx C ∧ A = C ∩ Y.
Apply iffI to the current goal.
Assume HAclosed: closed_in Y (subspace_topology X Tx Y) A.
We will prove ∃C : set, closed_in X Tx C ∧ A = C ∩ Y.
We prove the intermediate claim HTsubspace: topology_on Y (subspace_topology X Tx Y).
An exact proof term for the current goal is (subspace_topology_is_topology X Tx Y HTx HY).
We prove the intermediate claim HAdef: topology_on Y (subspace_topology X Tx Y) ∧ (A ⊆ Y ∧ ∃U ∈ subspace_topology X Tx Y, A = Y ∖ U).
An exact proof term for the current goal is HAclosed.
We prove the intermediate claim HAandEx: A ⊆ Y ∧ ∃U ∈ subspace_topology X Tx Y, A = Y ∖ U.
An exact proof term for the current goal is (andER (topology_on Y (subspace_topology X Tx Y)) (A ⊆ Y ∧ ∃U ∈ subspace_topology X Tx Y, A = Y ∖ U) HAdef).
We prove the intermediate claim HexU: ∃U ∈ subspace_topology X Tx Y, A = Y ∖ U.
An exact proof term for the current goal is (andER (A ⊆ Y) (∃U ∈ subspace_topology X Tx Y, A = Y ∖ U) HAandEx).
Apply HexU to the current goal.
Let U be given.
Assume HandEq.
Apply HandEq to the current goal.
Assume HUsubspace: U ∈ subspace_topology X Tx Y.
Assume HAeq: A = Y ∖ U.
We prove the intermediate claim HUexV: ∃V ∈ Tx, U = V ∩ Y.
An exact proof term for the current goal is (SepE2 (𝒫 Y) (λU0 : set ⇒ ∃V ∈ Tx, U0 = V ∩ Y) U HUsubspace).
Apply HUexV to the current goal.
Let V be given.
Assume HVandEq.
Apply HVandEq to the current goal.
Assume HV: V ∈ Tx.
Assume HUeq: U = V ∩ Y.
Set C to be the term X ∖ V.
We prove the intermediate claim HCclosed: closed_in X Tx C.
We will prove topology_on X Tx ∧ (C ⊆ X ∧ ∃W ∈ Tx, C = X ∖ W).
Apply andI to the current goal.
An exact proof term for the current goal is HTx.
Apply andI to the current goal.
An exact proof term for the current goal is (setminus_Subq X V).
We use V to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is HV.
Use reflexivity.
We prove the intermediate claim HAeqC: A = C ∩ Y.
rewrite the current goal using HAeq (from left to right).
rewrite the current goal using HUeq (from left to right).
We will prove Y ∖ (V ∩ Y) = (X ∖ V) ∩ Y.
Apply set_ext to the current goal.
Let x be given.
Assume Hx: x ∈ Y ∖ (V ∩ Y).
We prove the intermediate claim HxY: x ∈ Y.
An exact proof term for the current goal is (setminusE1 Y (V ∩ Y) x Hx).
We prove the intermediate claim HxnotVY: x ∉ V ∩ Y.
An exact proof term for the current goal is (setminusE2 Y (V ∩ Y) x Hx).
We prove the intermediate claim HxnotV: x ∉ V.
Assume HxV: x ∈ V.
Apply HxnotVY to the current goal.
Apply binintersectI to the current goal.
An exact proof term for the current goal is HxV.
An exact proof term for the current goal is HxY.
We prove the intermediate claim HxX: x ∈ X.
An exact proof term for the current goal is (HY x HxY).
Apply binintersectI to the current goal.
Apply setminusI to the current goal.
An exact proof term for the current goal is HxX.
An exact proof term for the current goal is HxnotV.
An exact proof term for the current goal is HxY.
Let x be given.
Assume Hx: x ∈ (X ∖ V) ∩ Y.
We prove the intermediate claim HxXV: x ∈ X ∖ V.
An exact proof term for the current goal is (binintersectE1 (X ∖ V) Y x Hx).
We prove the intermediate claim HxY: x ∈ Y.
An exact proof term for the current goal is (binintersectE2 (X ∖ V) Y x Hx).
We prove the intermediate claim HxnotV: x ∉ V.
An exact proof term for the current goal is (setminusE2 X V x HxXV).
Apply setminusI to the current goal.
An exact proof term for the current goal is HxY.
Assume HxVY: x ∈ V ∩ Y.
Apply HxnotV to the current goal.
An exact proof term for the current goal is (binintersectE1 V Y x HxVY).
We use C to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is HCclosed.
An exact proof term for the current goal is HAeqC.
Assume Hexists: ∃C : set, closed_in X Tx C ∧ A = C ∩ Y.
We will prove closed_in Y (subspace_topology X Tx Y) A.
Apply Hexists to the current goal.
Let C be given.
Assume HCandEq.
Apply HCandEq to the current goal.
Assume HCclosed: closed_in X Tx C.
Assume HAeq: A = C ∩ Y.
We prove the intermediate claim HTsubspace: topology_on Y (subspace_topology X Tx Y).
An exact proof term for the current goal is (subspace_topology_is_topology X Tx Y HTx HY).
We prove the intermediate claim HCdef: topology_on X Tx ∧ (C ⊆ X ∧ ∃V ∈ Tx, C = X ∖ V).
An exact proof term for the current goal is HCclosed.
We prove the intermediate claim HCandEx: C ⊆ X ∧ ∃V ∈ Tx, C = X ∖ V.
An exact proof term for the current goal is (andER (topology_on X Tx) (C ⊆ X ∧ ∃V ∈ Tx, C = X ∖ V) HCdef).
We prove the intermediate claim HexV: ∃V ∈ Tx, C = X ∖ V.
An exact proof term for the current goal is (andER (C ⊆ X) (∃V ∈ Tx, C = X ∖ V) HCandEx).
Apply HexV to the current goal.
Let V be given.
Assume HVandEq.
Apply HVandEq to the current goal.
Assume HV: V ∈ Tx.
Assume HCeq: C = X ∖ V.
Set U to be the term V ∩ Y.
We prove the intermediate claim HUsubspace: U ∈ subspace_topology X Tx Y.
We prove the intermediate claim HUinPowerY: U ∈ 𝒫 Y.
Apply PowerI to the current goal.
An exact proof term for the current goal is (binintersect_Subq_2 V Y).
We prove the intermediate claim HPred: ∃W ∈ Tx, U = W ∩ Y.
We use V to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is HV.
Use reflexivity.
An exact proof term for the current goal is (SepI (𝒫 Y) (λU0 : set ⇒ ∃W ∈ Tx, U0 = W ∩ Y) U HUinPowerY HPred).
We prove the intermediate claim HAeqYU: A = Y ∖ U.
rewrite the current goal using HAeq (from left to right).
rewrite the current goal using HCeq (from left to right).
We will prove (X ∖ V) ∩ Y = Y ∖ (V ∩ Y).
Apply set_ext to the current goal.
Let x be given.
Assume Hx: x ∈ (X ∖ V) ∩ Y.
We prove the intermediate claim HxXV: x ∈ X ∖ V.
An exact proof term for the current goal is (binintersectE1 (X ∖ V) Y x Hx).
We prove the intermediate claim HxY: x ∈ Y.
An exact proof term for the current goal is (binintersectE2 (X ∖ V) Y x Hx).
We prove the intermediate claim HxnotV: x ∉ V.
An exact proof term for the current goal is (setminusE2 X V x HxXV).
Apply setminusI to the current goal.
An exact proof term for the current goal is HxY.
Assume HxVY: x ∈ V ∩ Y.
Apply HxnotV to the current goal.
An exact proof term for the current goal is (binintersectE1 V Y x HxVY).
Let x be given.
Assume Hx: x ∈ Y ∖ (V ∩ Y).
We prove the intermediate claim HxY: x ∈ Y.
An exact proof term for the current goal is (setminusE1 Y (V ∩ Y) x Hx).
We prove the intermediate claim HxnotVY: x ∉ V ∩ Y.
An exact proof term for the current goal is (setminusE2 Y (V ∩ Y) x Hx).
We prove the intermediate claim HxnotV: x ∉ V.
Assume HxV: x ∈ V.
Apply HxnotVY to the current goal.
Apply binintersectI to the current goal.
An exact proof term for the current goal is HxV.
An exact proof term for the current goal is HxY.
We prove the intermediate claim HxX: x ∈ X.
An exact proof term for the current goal is (HY x HxY).
Apply binintersectI to the current goal.
Apply setminusI to the current goal.
An exact proof term for the current goal is HxX.
An exact proof term for the current goal is HxnotV.
An exact proof term for the current goal is HxY.
We prove the intermediate claim HAsub: A ⊆ Y.
rewrite the current goal using HAeq (from left to right).
An exact proof term for the current goal is (binintersect_Subq_2 C Y).
We will prove topology_on Y (subspace_topology X Tx Y) ∧ (A ⊆ Y ∧ ∃U ∈ subspace_topology X Tx Y, A = Y ∖ U).
Apply andI to the current goal.
An exact proof term for the current goal is HTsubspace.
Apply andI to the current goal.
An exact proof term for the current goal is HAsub.
We use U to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is HUsubspace.
An exact proof term for the current goal is HAeqYU.
∎
Theorem. (closed_in_closed_subspace)
∀X Tx Y A : set, topology_on X Tx → closed_in X Tx Y → closed_in Y (subspace_topology X Tx Y) A → closed_in X Tx A
In Proofgold the corresponding term root is 793cf4... and proposition id is 20f112...
Proof:
Let X, Tx, Y and A be given.
Assume HTx: topology_on X Tx.
Assume HY: closed_in X Tx Y.
Assume HA: closed_in Y (subspace_topology X Tx Y) A.
We will prove closed_in X Tx A.
We prove the intermediate claim HYsub: Y ⊆ X.
An exact proof term for the current goal is (andEL (Y ⊆ X) (∃U ∈ Tx, Y = X ∖ U) (andER (topology_on X Tx) (Y ⊆ X ∧ ∃U ∈ Tx, Y = X ∖ U) HY)).
We prove the intermediate claim Hexists: ∃C : set, closed_in X Tx C ∧ A = C ∩ Y.
Apply (iffEL (closed_in Y (subspace_topology X Tx Y) A) (∃C : set, closed_in X Tx C ∧ A = C ∩ Y) (closed_in_subspace_iff_intersection X Tx Y A HTx HYsub)) to the current goal.
An exact proof term for the current goal is HA.
Apply Hexists to the current goal.
Let C be given.
Assume HCandA: closed_in X Tx C ∧ A = C ∩ Y.
We prove the intermediate claim HCclosed: closed_in X Tx C.
An exact proof term for the current goal is (andEL (closed_in X Tx C) (A = C ∩ Y) HCandA).
We prove the intermediate claim HAeq: A = C ∩ Y.
An exact proof term for the current goal is (andER (closed_in X Tx C) (A = C ∩ Y) HCandA).
We prove the intermediate claim HCexists: ∃U ∈ Tx, C = X ∖ U.
An exact proof term for the current goal is (andER (C ⊆ X) (∃U ∈ Tx, C = X ∖ U) (andER (topology_on X Tx) (C ⊆ X ∧ ∃U ∈ Tx, C = X ∖ U) HCclosed)).
Apply HCexists to the current goal.
Let U be given.
Assume HU: U ∈ Tx ∧ C = X ∖ U.
We prove the intermediate claim HUinTx: U ∈ Tx.
An exact proof term for the current goal is (andEL (U ∈ Tx) (C = X ∖ U) HU).
We prove the intermediate claim HCeq: C = X ∖ U.
An exact proof term for the current goal is (andER (U ∈ Tx) (C = X ∖ U) HU).
We prove the intermediate claim HYexists: ∃V ∈ Tx, Y = X ∖ V.
An exact proof term for the current goal is (andER (Y ⊆ X) (∃V ∈ Tx, Y = X ∖ V) (andER (topology_on X Tx) (Y ⊆ X ∧ ∃V ∈ Tx, Y = X ∖ V) HY)).
Apply HYexists to the current goal.
Let V be given.
Assume HV: V ∈ Tx ∧ Y = X ∖ V.
We prove the intermediate claim HVinTx: V ∈ Tx.
An exact proof term for the current goal is (andEL (V ∈ Tx) (Y = X ∖ V) HV).
We prove the intermediate claim HYeq: Y = X ∖ V.
An exact proof term for the current goal is (andER (V ∈ Tx) (Y = X ∖ V) HV).
We prove the intermediate claim HAeqSetminus: A = (X ∖ U) ∩ (X ∖ V).
rewrite the current goal using HAeq (from left to right).
rewrite the current goal using HCeq (from left to right).
rewrite the current goal using HYeq (from left to right).
Use reflexivity.
We prove the intermediate claim HDeM: (X ∖ U) ∩ (X ∖ V) = X ∖ (U ∪ V).
Apply set_ext to the current goal.
Let x be given.
Assume Hx: x ∈ (X ∖ U) ∩ (X ∖ V).
We will prove x ∈ X ∖ (U ∪ V).
We prove the intermediate claim HxXU: x ∈ X ∖ U.
An exact proof term for the current goal is (binintersectE1 (X ∖ U) (X ∖ V) x Hx).
We prove the intermediate claim HxXV: x ∈ X ∖ V.
An exact proof term for the current goal is (binintersectE2 (X ∖ U) (X ∖ V) x Hx).
We prove the intermediate claim HxX: x ∈ X.
An exact proof term for the current goal is (setminusE1 X U x HxXU).
We prove the intermediate claim HxninU: x ∉ U.
An exact proof term for the current goal is (setminusE2 X U x HxXU).
We prove the intermediate claim HxninV: x ∉ V.
An exact proof term for the current goal is (setminusE2 X V x HxXV).
Apply setminusI to the current goal.
An exact proof term for the current goal is HxX.
Assume HxUV: x ∈ U ∪ V.
We will prove False.
Apply (binunionE U V x HxUV) to the current goal.
Assume HxU: x ∈ U.
An exact proof term for the current goal is (HxninU HxU).
Assume HxV: x ∈ V.
An exact proof term for the current goal is (HxninV HxV).
Let x be given.
Assume Hx: x ∈ X ∖ (U ∪ V).
We will prove x ∈ (X ∖ U) ∩ (X ∖ V).
We prove the intermediate claim HxX: x ∈ X.
An exact proof term for the current goal is (setminusE1 X (U ∪ V) x Hx).
We prove the intermediate claim HxninUV: x ∉ U ∪ V.
An exact proof term for the current goal is (setminusE2 X (U ∪ V) x Hx).
Apply binintersectI to the current goal.
We will prove x ∈ X ∖ U.
Apply setminusI to the current goal.
An exact proof term for the current goal is HxX.
Assume HxU: x ∈ U.
We will prove False.
We prove the intermediate claim HxUV: x ∈ U ∪ V.
An exact proof term for the current goal is (binunionI1 U V x HxU).
An exact proof term for the current goal is (HxninUV HxUV).
We will prove x ∈ X ∖ V.
Apply setminusI to the current goal.
An exact proof term for the current goal is HxX.
Assume HxV: x ∈ V.
We will prove False.
We prove the intermediate claim HxUV: x ∈ U ∪ V.
An exact proof term for the current goal is (binunionI2 U V x HxV).
An exact proof term for the current goal is (HxninUV HxUV).
We prove the intermediate claim HUV: U ∪ V ∈ Tx.
We prove the intermediate claim HUV_eq: U ∪ V = ⋃ (UPair U V).
Apply set_ext to the current goal.
Let x be given.
Assume Hx: x ∈ U ∪ V.
We will prove x ∈ ⋃ (UPair U V).
Apply (binunionE U V x Hx) to the current goal.
Assume HxU: x ∈ U.
Apply (UnionI (UPair U V) x U HxU) to the current goal.
An exact proof term for the current goal is (UPairI1 U V).
Assume HxV: x ∈ V.
Apply (UnionI (UPair U V) x V HxV) to the current goal.
An exact proof term for the current goal is (UPairI2 U V).
Let x be given.
Assume Hx: x ∈ ⋃ (UPair U V).
We will prove x ∈ U ∪ V.
Apply (UnionE_impred (UPair U V) x Hx (x ∈ U ∪ V)) to the current goal.
Let W be given.
Assume HxW: x ∈ W.
Assume HWin: W ∈ UPair U V.
Apply (UPairE W U V HWin) to the current goal.
Assume HWeqU: W = U.
We prove the intermediate claim HxU: x ∈ U.
rewrite the current goal using HWeqU (from right to left).
An exact proof term for the current goal is HxW.
An exact proof term for the current goal is (binunionI1 U V x HxU).
Assume HWeqV: W = V.
We prove the intermediate claim HxV: x ∈ V.
rewrite the current goal using HWeqV (from right to left).
An exact proof term for the current goal is HxW.
An exact proof term for the current goal is (binunionI2 U V x HxV).
rewrite the current goal using HUV_eq (from left to right).
We prove the intermediate claim HUPairSub: UPair U V ⊆ Tx.
Let W be given.
Assume HW: W ∈ UPair U V.
We will prove W ∈ Tx.
Apply (UPairE W U V HW) to the current goal.
Assume HWeqU: W = U.
rewrite the current goal using HWeqU (from left to right).
An exact proof term for the current goal is HUinTx.
Assume HWeqV: W = V.
rewrite the current goal using HWeqV (from left to right).
An exact proof term for the current goal is HVinTx.
An exact proof term for the current goal is (topology_union_closed X Tx (UPair U V) HTx HUPairSub).
We prove the intermediate claim HAeqFinal: A = X ∖ (U ∪ V).
rewrite the current goal using HAeqSetminus (from left to right).
An exact proof term for the current goal is HDeM.
We will prove topology_on X Tx ∧ (A ⊆ X ∧ ∃U0 ∈ Tx, A = X ∖ U0).
Apply andI to the current goal.
An exact proof term for the current goal is HTx.
Apply andI to the current goal.
We will prove A ⊆ X.
rewrite the current goal using HAeqFinal (from left to right).
An exact proof term for the current goal is (setminus_Subq X (U ∪ V)).
We will prove ∃U0 ∈ Tx, A = X ∖ U0.
We use (U ∪ V) to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is HUV.
An exact proof term for the current goal is HAeqFinal.
∎
Theorem. (closure_in_subspace)
∀X Tx Y A : set, topology_on X Tx → Y ⊆ X → closure_of Y (subspace_topology X Tx Y) A = (closure_of X Tx A) ∩ Y
In Proofgold the corresponding term root is 95a861... and proposition id is 0e4f16...
Proof:
Let X, Tx, Y and A be given.
Assume HTx: topology_on X Tx.
Assume HY: Y ⊆ X.
We will prove closure_of Y (subspace_topology X Tx Y) A = (closure_of X Tx A) ∩ Y.
We prove the intermediate claim HTy: topology_on Y (subspace_topology X Tx Y).
An exact proof term for the current goal is (subspace_topology_is_topology X Tx Y HTx HY).
Apply set_ext to the current goal.
Let y be given.
Assume Hy: y ∈ closure_of Y (subspace_topology X Tx Y) A.
We will prove y ∈ (closure_of X Tx A) ∩ Y.
We prove the intermediate claim HyY: y ∈ Y.
An exact proof term for the current goal is (SepE1 Y (λy0 ⇒ ∀U : set, U ∈ subspace_topology X Tx Y → y0 ∈ U → U ∩ A ≠ Empty) y Hy).
We prove the intermediate claim HysubCond: ∀U : set, U ∈ subspace_topology X Tx Y → y ∈ U → U ∩ A ≠ Empty.
An exact proof term for the current goal is (SepE2 Y (λy0 ⇒ ∀U : set, U ∈ subspace_topology X Tx Y → y0 ∈ U → U ∩ A ≠ Empty) y Hy).
Apply binintersectI to the current goal.
We will prove y ∈ closure_of X Tx A.
We prove the intermediate claim HyX: y ∈ X.
An exact proof term for the current goal is (HY y HyY).
We prove the intermediate claim HyCond: ∀V : set, V ∈ Tx → y ∈ V → V ∩ A ≠ Empty.
Let V be given.
Assume HV: V ∈ Tx.
Assume HyV: y ∈ V.
We will prove V ∩ A ≠ Empty.
Set U to be the term V ∩ Y.
We prove the intermediate claim HU: U ∈ subspace_topology X Tx Y.
We prove the intermediate claim HUinPower: U ∈ 𝒫 Y.
Apply PowerI to the current goal.
An exact proof term for the current goal is (binintersect_Subq_2 V Y).
We prove the intermediate claim HPred: ∃W ∈ Tx, U = W ∩ Y.
We use V to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is HV.
Use reflexivity.
An exact proof term for the current goal is (SepI (𝒫 Y) (λU0 ⇒ ∃W ∈ Tx, U0 = W ∩ Y) U HUinPower HPred).
We prove the intermediate claim HyU: y ∈ U.
Apply binintersectI to the current goal.
An exact proof term for the current goal is HyV.
An exact proof term for the current goal is HyY.
We prove the intermediate claim HUA_ne: U ∩ A ≠ Empty.
An exact proof term for the current goal is (HysubCond U HU HyU).
We will prove V ∩ A ≠ Empty.
Assume HVA_empty: V ∩ A = Empty.
We prove the intermediate claim HUA_sub_VA: U ∩ A ⊆ V ∩ A.
Let z be given.
Assume Hz: z ∈ U ∩ A.
We prove the intermediate claim HzU: z ∈ U.
An exact proof term for the current goal is (binintersectE1 U A z Hz).
We prove the intermediate claim HzV: z ∈ V.
An exact proof term for the current goal is (binintersectE1 V Y z HzU).
We prove the intermediate claim HzA: z ∈ A.
An exact proof term for the current goal is (binintersectE2 U A z Hz).
An exact proof term for the current goal is (binintersectI V A z HzV HzA).
We prove the intermediate claim HUA_sub_Empty: U ∩ A ⊆ Empty.
rewrite the current goal using HVA_empty (from right to left).
An exact proof term for the current goal is HUA_sub_VA.
We prove the intermediate claim HUA_empty: U ∩ A = Empty.
An exact proof term for the current goal is (Empty_Subq_eq (U ∩ A) HUA_sub_Empty).
An exact proof term for the current goal is (HUA_ne HUA_empty).
An exact proof term for the current goal is (SepI X (λy0 ⇒ ∀V : set, V ∈ Tx → y0 ∈ V → V ∩ A ≠ Empty) y HyX HyCond).
An exact proof term for the current goal is HyY.
Let y be given.
Assume Hy: y ∈ (closure_of X Tx A) ∩ Y.
We will prove y ∈ closure_of Y (subspace_topology X Tx Y) A.
We prove the intermediate claim HyClX: y ∈ closure_of X Tx A.
An exact proof term for the current goal is (binintersectE1 (closure_of X Tx A) Y y Hy).
We prove the intermediate claim HyY: y ∈ Y.
An exact proof term for the current goal is (binintersectE2 (closure_of X Tx A) Y y Hy).
We prove the intermediate claim HyXCond: ∀V : set, V ∈ Tx → y ∈ V → V ∩ A ≠ Empty.
An exact proof term for the current goal is (SepE2 X (λy0 ⇒ ∀V : set, V ∈ Tx → y0 ∈ V → V ∩ A ≠ Empty) y HyClX).
We prove the intermediate claim HySubCond: ∀U : set, U ∈ subspace_topology X Tx Y → y ∈ U → U ∩ A ≠ Empty.
Let U be given.
Assume HU: U ∈ subspace_topology X Tx Y.
Assume HyU: y ∈ U.
We will prove U ∩ A ≠ Empty.
We prove the intermediate claim HUex: ∃V ∈ Tx, U = V ∩ Y.
An exact proof term for the current goal is (SepE2 (𝒫 Y) (λU0 ⇒ ∃V ∈ Tx, U0 = V ∩ Y) U HU).
Apply HUex to the current goal.
Let V be given.
Assume HVandEq.
Apply HVandEq to the current goal.
Assume HV: V ∈ Tx.
Assume HUeq: U = V ∩ Y.
We prove the intermediate claim HyV: y ∈ V.
We prove the intermediate claim HyVY: y ∈ V ∩ Y.
rewrite the current goal using HUeq (from right to left).
An exact proof term for the current goal is HyU.
An exact proof term for the current goal is (binintersectE1 V Y y HyVY).
We prove the intermediate claim HVA_ne: V ∩ A ≠ Empty.
An exact proof term for the current goal is (HyXCond V HV HyV).
rewrite the current goal using HUeq (from left to right).
We will prove (V ∩ Y) ∩ A ≠ Empty.
Assume HVYAempty: (V ∩ Y) ∩ A = Empty.
We prove the intermediate claim HVA_sub_VYA: V ∩ (Y ∩ A) ⊆ V ∩ A.
Let z be given.
Assume Hz: z ∈ V ∩ (Y ∩ A).
We prove the intermediate claim HzV: z ∈ V.
An exact proof term for the current goal is (binintersectE1 V (Y ∩ A) z Hz).
We prove the intermediate claim HzYA: z ∈ Y ∩ A.
An exact proof term for the current goal is (binintersectE2 V (Y ∩ A) z Hz).
We prove the intermediate claim HzA: z ∈ A.
An exact proof term for the current goal is (binintersectE2 Y A z HzYA).
An exact proof term for the current goal is (binintersectI V A z HzV HzA).
We prove the intermediate claim HVYAeq: V ∩ (Y ∩ A) = (V ∩ Y) ∩ A.
Apply set_ext to the current goal.
Let z be given.
Assume Hz: z ∈ V ∩ (Y ∩ A).
We prove the intermediate claim HzV: z ∈ V.
An exact proof term for the current goal is (binintersectE1 V (Y ∩ A) z Hz).
We prove the intermediate claim HzYA: z ∈ Y ∩ A.
An exact proof term for the current goal is (binintersectE2 V (Y ∩ A) z Hz).
We prove the intermediate claim HzY: z ∈ Y.
An exact proof term for the current goal is (binintersectE1 Y A z HzYA).
We prove the intermediate claim HzA: z ∈ A.
An exact proof term for the current goal is (binintersectE2 Y A z HzYA).
We prove the intermediate claim HzVY: z ∈ V ∩ Y.
An exact proof term for the current goal is (binintersectI V Y z HzV HzY).
An exact proof term for the current goal is (binintersectI (V ∩ Y) A z HzVY HzA).
Let z be given.
Assume Hz: z ∈ (V ∩ Y) ∩ A.
We prove the intermediate claim HzVY: z ∈ V ∩ Y.
An exact proof term for the current goal is (binintersectE1 (V ∩ Y) A z Hz).
We prove the intermediate claim HzV: z ∈ V.
An exact proof term for the current goal is (binintersectE1 V Y z HzVY).
We prove the intermediate claim HzY: z ∈ Y.
An exact proof term for the current goal is (binintersectE2 V Y z HzVY).
We prove the intermediate claim HzA: z ∈ A.
An exact proof term for the current goal is (binintersectE2 (V ∩ Y) A z Hz).
We prove the intermediate claim HzYA: z ∈ Y ∩ A.
An exact proof term for the current goal is (binintersectI Y A z HzY HzA).
An exact proof term for the current goal is (binintersectI V (Y ∩ A) z HzV HzYA).
We prove the intermediate claim HVYAempty2: V ∩ (Y ∩ A) = Empty.
rewrite the current goal using HVYAeq (from left to right).
An exact proof term for the current goal is HVYAempty.
The rest of this subproof is missing.
An exact proof term for the current goal is (SepI Y (λy0 ⇒ ∀U : set, U ∈ subspace_topology X Tx Y → y0 ∈ U → U ∩ A ≠ Empty) y HyY HySubCond).
∎
Theorem. (closure_characterization)
∀X Tx A x : set, topology_on X Tx → (x ∈ closure_of X Tx A ↔ (∀U ∈ Tx, x ∈ U → U ∩ A ≠ Empty))
In Proofgold the corresponding term root is 8b92b4... and proposition id is c1acfc...
Proof:
Let X, Tx, A and x be given.
Assume HTx: topology_on X Tx.
We will prove x ∈ closure_of X Tx A ↔ (∀U ∈ Tx, x ∈ U → U ∩ A ≠ Empty).
Apply iffI to the current goal.
Assume Hx: x ∈ closure_of X Tx A.
We will prove ∀U ∈ Tx, x ∈ U → U ∩ A ≠ Empty.
An exact proof term for the current goal is (SepE2 X (λx0 ⇒ ∀U : set, U ∈ Tx → x0 ∈ U → U ∩ A ≠ Empty) x Hx).
Assume Hcond: ∀U ∈ Tx, x ∈ U → U ∩ A ≠ Empty.
We will prove x ∈ closure_of X Tx A.
We prove the intermediate claim HXinTx: X ∈ Tx.
An exact proof term for the current goal is (topology_has_X X Tx HTx).
Apply xm (x ∈ X) to the current goal.
Assume HxX: x ∈ X.
An exact proof term for the current goal is (SepI X (λx0 ⇒ ∀U : set, U ∈ Tx → x0 ∈ U → U ∩ A ≠ Empty) x HxX Hcond).
Assume HxnotX: x ∉ X.
Apply FalseE to the current goal.
The rest of this subproof is missing.
∎
Definition. We define limit_point_of to be λX Tx A x ⇒ topology_on X Tx ∧ x ∈ X ∧ ∀U : set, U ∈ Tx → x ∈ U → ∃y : set, y ∈ A ∧ y ≠ x ∧ y ∈ U of type set → set → set → set → prop.
In Proofgold the corresponding term root is 05a110... and object id is 456f0c...
Definition. We define limit_points_of to be λX Tx A ⇒ {x ∈ X|limit_point_of X Tx A x} of type set → set → set → set.
In Proofgold the corresponding term root is d2aeb4... and object id is 31b1d2...
Theorem. (closure_equals_set_plus_limit_points)
∀X Tx A : set, topology_on X Tx → closure_of X Tx A = A ∪ limit_points_of X Tx A
In Proofgold the corresponding term root is f63d48... and proposition id is 4e173a...
Proof:
Let X, Tx and A be given.
Assume HTx: topology_on X Tx.
We will prove closure_of X Tx A = A ∪ limit_points_of X Tx A.
The rest of this subproof is missing.
∎
Theorem. (closed_iff_contains_limit_points)
∀X Tx A : set, topology_on X Tx → (closed_in X Tx A ↔ limit_points_of X Tx A ⊆ A)
In Proofgold the corresponding term root is 78f8ea... and proposition id is 0c75c0...
Proof:
Let X, Tx and A be given.
Assume HTx: topology_on X Tx.
We will prove closed_in X Tx A ↔ limit_points_of X Tx A ⊆ A.
The rest of this subproof is missing.
∎
Definition. We define Hausdorff_space to be λX Tx ⇒ topology_on X Tx ∧ ∀x1 x2 : set, x1 ≠ x2 → ∃U V : set, U ∈ Tx ∧ V ∈ Tx ∧ x1 ∈ U ∧ x2 ∈ V ∧ U ∩ V = Empty of type set → set → prop.
In Proofgold the corresponding term root is 2e67c7... and object id is ec0214...
Definition. We define T1_space to be λX Tx ⇒ topology_on X Tx ∧ (∀F : set, finite F → closed_in X Tx F) of type set → set → prop.
In Proofgold the corresponding term root is e5c23f... and object id is 1c6229...
Theorem. (finite_sets_closed_in_Hausdorff)
∀X Tx : set, Hausdorff_space X Tx → ∀F : set, finite F → closed_in X Tx F
In Proofgold the corresponding term root is 34b358... and proposition id is 9ce212...
Proof:
Let X and Tx be given.
Assume HH: Hausdorff_space X Tx.
Let F be given.
Assume HF: finite F.
We will prove closed_in X Tx F.
The rest of this subproof is missing.
∎
Theorem. (limit_points_infinite_neighborhoods)
∀X Tx A x : set, T1_space X Tx → (limit_point_of X Tx A x ↔ (∀U ∈ Tx, x ∈ U → infinite (U ∩ A)))
In Proofgold the corresponding term root is b3fbaa... and proposition id is 379043...
Proof:
Let X, Tx, A and x be given.
Assume HT1: T1_space X Tx.
We will prove limit_point_of X Tx A x ↔ (∀U ∈ Tx, x ∈ U → infinite (U ∩ A)).
The rest of this subproof is missing.
∎
Theorem. (Hausdorff_unique_limits)
∀X Tx seq x y : set, Hausdorff_space X Tx → x ≠ y → function_on seq ω X → (∀U : set, U ∈ Tx → x ∈ U → ∃N : set, N ∈ ω ∧ ∀n : set, n ∈ ω → N ⊆ n → apply_fun seq n ∈ U) → (∀U : set, U ∈ Tx → y ∈ U → ∃N : set, N ∈ ω ∧ ∀n : set, n ∈ ω → N ⊆ n → apply_fun seq n ∈ U) → False
In Proofgold the corresponding term root is 985c41... and proposition id is 41b4c6...
Proof:
Let X, Tx, seq, x and y be given.
Assume HH: Hausdorff_space X Tx.
Assume Hneq: x ≠ y.
Assume Hseq: function_on seq ω X.
Assume Hx: ∀U : set, U ∈ Tx → x ∈ U → ∃N : set, N ∈ ω ∧ ∀n : set, n ∈ ω → N ⊆ n → apply_fun seq n ∈ U.
Assume Hy: ∀U : set, U ∈ Tx → y ∈ U → ∃N : set, N ∈ ω ∧ ∀n : set, n ∈ ω → N ⊆ n → apply_fun seq n ∈ U.
We will prove False.
The rest of this subproof is missing.
∎
Theorem. (Hausdorff_stability)
∀X Tx Y Ty : set, Hausdorff_space X Tx ∧ Hausdorff_space Y Ty → Hausdorff_space (OrderedPair X Y) (product_topology X Tx Y Ty)
In Proofgold the corresponding term root is 2b294e... and proposition id is 89be76...
Proof:
Let X, Tx, Y and Ty be given.
Assume H: Hausdorff_space X Tx ∧ Hausdorff_space Y Ty.
We will prove Hausdorff_space (OrderedPair X Y) (product_topology X Tx Y Ty).
The rest of this subproof is missing.
∎
Theorem. (ex17_1_topology_from_closed_sets)
∀X Tx : set, closed_in X Tx X → (∀A : set, closed_in X Tx A → closed_in X Tx (X ∖ A)) → topology_on X Tx
In Proofgold the corresponding term root is 380cd0... and proposition id is b87d64...
Proof:
Let X and Tx be given.
Assume H1: closed_in X Tx X.
Assume H2: ∀A : set, closed_in X Tx A → closed_in X Tx (X ∖ A).
We will prove topology_on X Tx.
The rest of this subproof is missing.
∎
Theorem. (ex17_2_closed_in_closed_subspace)
∀X Tx Y A : set, closed_in X Tx Y → closed_in Y (subspace_topology X Tx Y) A → closed_in X Tx A
In Proofgold the corresponding term root is f25458... and proposition id is acc8de...
Proof:
Let X, Tx, Y and A be given.
Assume HY: closed_in X Tx Y.
Assume HA: closed_in Y (subspace_topology X Tx Y) A.
We will prove closed_in X Tx A.
We prove the intermediate claim HTx: topology_on X Tx.
An exact proof term for the current goal is (andEL (topology_on X Tx) (Y ⊆ X ∧ ∃U ∈ Tx, Y = X ∖ U) HY).
An exact proof term for the current goal is (closed_in_closed_subspace X Tx Y A HTx HY HA).
∎
Theorem. (ex17_3_product_of_closed_sets_closed)
∀X Tx Y Ty A B : set, closed_in X Tx A → closed_in Y Ty B → closed_in (OrderedPair X Y) (product_topology X Tx Y Ty) (OrderedPair A B)
In Proofgold the corresponding term root is 5a8115... and proposition id is 39ae31...
Proof:
Let X, Tx, Y, Ty, A and B be given.
Assume HA: closed_in X Tx A.
Assume HB: closed_in Y Ty B.
We will prove closed_in (OrderedPair X Y) (product_topology X Tx Y Ty) (OrderedPair A B).
The rest of this subproof is missing.
∎
Theorem. (ex17_4_open_minus_closed_and_closed_minus_open)
∀X Tx U A : set, topology_on X Tx → open_in X Tx U → closed_in X Tx A → open_in X Tx (U ∖ A) ∧ closed_in X Tx (A ∖ U)
In Proofgold the corresponding term root is 80566d... and proposition id is 091281...
Proof:
Let X, Tx, U and A be given.
Assume Htop: topology_on X Tx.
Assume HU: open_in X Tx U.
Assume HA: closed_in X Tx A.
We will prove open_in X Tx (U ∖ A) ∧ closed_in X Tx (A ∖ U).
We prove the intermediate claim HUtop: U ∈ Tx.
An exact proof term for the current goal is (andER (topology_on X Tx) (U ∈ Tx) HU).
We prove the intermediate claim HAdef: A ⊆ X ∧ (∃V ∈ Tx, A = X ∖ V).
An exact proof term for the current goal is (andER (topology_on X Tx) (A ⊆ X ∧ (∃V ∈ Tx, A = X ∖ V)) HA).
We prove the intermediate claim HexV: ∃V ∈ Tx, A = X ∖ V.
An exact proof term for the current goal is (andER (A ⊆ X) (∃V ∈ Tx, A = X ∖ V) HAdef).
Apply HexV to the current goal.
Let V be given.
Assume HVandEq.
Apply HVandEq to the current goal.
Assume HV: V ∈ Tx.
Assume HAeq: A = X ∖ V.
Apply andI to the current goal.
We will prove open_in X Tx (U ∖ A).
We prove the intermediate claim HUsub: U ⊆ X.
An exact proof term for the current goal is (topology_elem_subset X Tx U Htop HUtop).
We prove the intermediate claim HUminusA_eq_UinterV: U ∖ A = U ∩ V.
rewrite the current goal using HAeq (from left to right).
Apply set_ext to the current goal.
Let x be given.
Assume Hx: x ∈ U ∖ (X ∖ V).
We prove the intermediate claim HxU: x ∈ U.
An exact proof term for the current goal is (setminusE1 U (X ∖ V) x Hx).
We prove the intermediate claim HxnotXV: x ∉ X ∖ V.
An exact proof term for the current goal is (setminusE2 U (X ∖ V) x Hx).
We prove the intermediate claim HxV: x ∈ V.
We prove the intermediate claim HxX: x ∈ X.
An exact proof term for the current goal is (HUsub x HxU).
Apply xm (x ∈ V) to the current goal.
Assume Hv.
An exact proof term for the current goal is Hv.
Assume Hnv.
Apply FalseE to the current goal.
Apply HxnotXV to the current goal.
An exact proof term for the current goal is (setminusI X V x HxX Hnv).
An exact proof term for the current goal is (binintersectI U V x HxU HxV).
Let x be given.
Assume Hx: x ∈ U ∩ V.
We prove the intermediate claim HxU: x ∈ U.
An exact proof term for the current goal is (binintersectE1 U V x Hx).
We prove the intermediate claim HxV: x ∈ V.
An exact proof term for the current goal is (binintersectE2 U V x Hx).
We prove the intermediate claim HxnotXV: x ∉ X ∖ V.
Assume H.
Apply (setminusE2 X V x H) to the current goal.
An exact proof term for the current goal is HxV.
An exact proof term for the current goal is (setminusI U (X ∖ V) x HxU HxnotXV).
rewrite the current goal using HUminusA_eq_UinterV (from left to right).
We prove the intermediate claim HUinterV: U ∩ V ∈ Tx.
An exact proof term for the current goal is (topology_binintersect_closed X Tx U V Htop HUtop HV).
An exact proof term for the current goal is (andI (topology_on X Tx) (U ∩ V ∈ Tx) Htop HUinterV).
We will prove closed_in X Tx (A ∖ U).
We prove the intermediate claim HAsub: A ⊆ X.
An exact proof term for the current goal is (andEL (A ⊆ X) (∃V0 ∈ Tx, A = X ∖ V0) HAdef).
We prove the intermediate claim HAminusU_sub: A ∖ U ⊆ X.
Let x be given.
Assume Hx.
We prove the intermediate claim HxA: x ∈ A.
An exact proof term for the current goal is (setminusE1 A U x Hx).
An exact proof term for the current goal is (HAsub x HxA).
We prove the intermediate claim HVU: V ∪ U ∈ Tx.
Set UFam to be the term UPair V U.
We prove the intermediate claim HUFamsub: UFam ⊆ Tx.
Let W be given.
Assume HW: W ∈ UFam.
Apply (UPairE W V U HW) to the current goal.
Assume HWeqV.
rewrite the current goal using HWeqV (from left to right).
An exact proof term for the current goal is HV.
Assume HWeqU.
rewrite the current goal using HWeqU (from left to right).
An exact proof term for the current goal is HUtop.
We prove the intermediate claim HUnionVU: ⋃ UFam = V ∪ U.
Apply set_ext to the current goal.
Let x be given.
Assume Hx: x ∈ ⋃ UFam.
Apply UnionE_impred UFam x Hx to the current goal.
Let W be given.
Assume HxW: x ∈ W.
Assume HW: W ∈ UFam.
Apply (UPairE W V U HW) to the current goal.
Assume HWeqV.
We prove the intermediate claim HxV: x ∈ V.
rewrite the current goal using HWeqV (from right to left).
An exact proof term for the current goal is HxW.
An exact proof term for the current goal is (binunionI1 V U x HxV).
Assume HWeqU.
We prove the intermediate claim HxU: x ∈ U.
rewrite the current goal using HWeqU (from right to left).
An exact proof term for the current goal is HxW.
An exact proof term for the current goal is (binunionI2 V U x HxU).
Let x be given.
Assume Hx: x ∈ V ∪ U.
Apply (binunionE V U x Hx) to the current goal.
Assume HxV.
An exact proof term for the current goal is (UnionI UFam x V HxV (UPairI1 V U)).
Assume HxU.
An exact proof term for the current goal is (UnionI UFam x U HxU (UPairI2 V U)).
rewrite the current goal using HUnionVU (from right to left).
An exact proof term for the current goal is (topology_union_closed X Tx UFam Htop HUFamsub).
We prove the intermediate claim HAminusU_eq_XminusVU: A ∖ U = X ∖ (V ∪ U).
rewrite the current goal using HAeq (from left to right).
Apply set_ext to the current goal.
Let x be given.
Assume Hx: x ∈ (X ∖ V) ∖ U.
We prove the intermediate claim HxXV: x ∈ X ∖ V.
An exact proof term for the current goal is (setminusE1 (X ∖ V) U x Hx).
We prove the intermediate claim HxnotU: x ∉ U.
An exact proof term for the current goal is (setminusE2 (X ∖ V) U x Hx).
We prove the intermediate claim HxX: x ∈ X.
An exact proof term for the current goal is (setminusE1 X V x HxXV).
We prove the intermediate claim HxnotV: x ∉ V.
An exact proof term for the current goal is (setminusE2 X V x HxXV).
We prove the intermediate claim HxnotVU: x ∉ V ∪ U.
Assume H.
Apply (binunionE V U x H) to the current goal.
Assume HxV.
An exact proof term for the current goal is (HxnotV HxV).
Assume HxU.
An exact proof term for the current goal is (HxnotU HxU).
An exact proof term for the current goal is (setminusI X (V ∪ U) x HxX HxnotVU).
Let x be given.
Assume Hx: x ∈ X ∖ (V ∪ U).
We prove the intermediate claim HxX: x ∈ X.
An exact proof term for the current goal is (setminusE1 X (V ∪ U) x Hx).
We prove the intermediate claim HxnotVU: x ∉ V ∪ U.
An exact proof term for the current goal is (setminusE2 X (V ∪ U) x Hx).
We prove the intermediate claim HxnotV: x ∉ V.
Assume HxV.
Apply HxnotVU to the current goal.
An exact proof term for the current goal is (binunionI1 V U x HxV).
We prove the intermediate claim HxnotU: x ∉ U.
Assume HxU.
Apply HxnotVU to the current goal.
An exact proof term for the current goal is (binunionI2 V U x HxU).
We prove the intermediate claim HxXV: x ∈ X ∖ V.
An exact proof term for the current goal is (setminusI X V x HxX HxnotV).
An exact proof term for the current goal is (setminusI (X ∖ V) U x HxXV HxnotU).
We prove the intermediate claim HPred: ∃W ∈ Tx, A ∖ U = X ∖ W.
We use (V ∪ U) to witness the existential quantifier.
Apply andI to the current goal.
An exact proof term for the current goal is HVU.
An exact proof term for the current goal is HAminusU_eq_XminusVU.
An exact proof term for the current goal is (andI (topology_on X Tx) (A ∖ U ⊆ X ∧ (∃W ∈ Tx, A ∖ U = X ∖ W)) Htop (andI (A ∖ U ⊆ X) (∃W ∈ Tx, A ∖ U = X ∖ W) HAminusU_sub HPred)).
∎
Theorem. (ex17_5_closure_of_interval_in_order_topology)
∀X : set, closure_of X (order_topology X) (open_interval 0 1) = open_interval 0 1
In Proofgold the corresponding term root is 0c2ef9... and proposition id is 65c3b7...
Proof:
Let X be given.
We will prove closure_of X (order_topology X) (open_interval 0 1) = open_interval 0 1.
The rest of this subproof is missing.
∎
Theorem. (ex17_6_closure_properties)
∀X Tx A : set, topology_on X Tx → closure_of X Tx (closure_of X Tx A) = closure_of X Tx A ∧ closed_in X Tx (closure_of X Tx A)
In Proofgold the corresponding term root is 9b66b0... and proposition id is 967e99...
Proof:
Let X, Tx and A be given.
Assume Htop: topology_on X Tx.
We will prove closure_of X Tx (closure_of X Tx A) = closure_of X Tx A ∧ closed_in X Tx (closure_of X Tx A).
Apply andI to the current goal.
We will prove closure_of X Tx (closure_of X Tx A) = closure_of X Tx A.
The rest of this subproof is missing.
We will prove closed_in X Tx (closure_of X Tx A).
The rest of this subproof is missing.
∎
Theorem. (ex17_7_counterexample_union_closure)
∀X Tx A B : set, topology_on X Tx → closed_in X Tx (A ∪ B) → ¬ (closed_in X Tx A ∧ closed_in X Tx B)
In Proofgold the corresponding term root is 855cc8... and proposition id is c7f726...
Proof:
Let X, Tx, A and B be given.
Assume Htop: topology_on X Tx.
Assume HAB: closed_in X Tx (A ∪ B).
We will prove ¬ (closed_in X Tx A ∧ closed_in X Tx B).
The rest of this subproof is missing.
∎
Theorem. (ex17_8_closure_intersection_questions)
∀X Tx A B : set, topology_on X Tx → closure_of X Tx (A ∩ B) ⊆ closure_of X Tx A ∩ closure_of X Tx B
In Proofgold the corresponding term root is c96078... and proposition id is a800db...
Proof:
Let X, Tx, A and B be given.
Assume Htop: topology_on X Tx.
We will prove closure_of X Tx (A ∩ B) ⊆ closure_of X Tx A ∩ closure_of X Tx B.
Let x be given.
Assume Hx: x ∈ closure_of X Tx (A ∩ B).
We will prove x ∈ closure_of X Tx A ∩ closure_of X Tx B.
We prove the intermediate claim HxX: x ∈ X.
An exact proof term for the current goal is (SepE1 X (λx0 ⇒ ∀U : set, U ∈ Tx → x0 ∈ U → U ∩ (A ∩ B) ≠ Empty) x Hx).
We prove the intermediate claim HxAB: ∀U : set, U ∈ Tx → x ∈ U → U ∩ (A ∩ B) ≠ Empty.
An exact proof term for the current goal is (SepE2 X (λx0 ⇒ ∀U : set, U ∈ Tx → x0 ∈ U → U ∩ (A ∩ B) ≠ Empty) x Hx).
Apply binintersectI to the current goal.
We will prove x ∈ closure_of X Tx A.
We prove the intermediate claim HxA: x ∈ X ∧ (∀U : set, U ∈ Tx → x ∈ U → U ∩ A ≠ Empty).
Apply andI to the current goal.
An exact proof term for the current goal is HxX.
Let U be given.
Assume HU: U ∈ Tx.
Assume HxU: x ∈ U.
We will prove U ∩ A ≠ Empty.
We prove the intermediate claim HABne: U ∩ (A ∩ B) ≠ Empty.
An exact proof term for the current goal is (HxAB U HU HxU).
Assume Hempty: U ∩ A = Empty.
Apply HABne to the current goal.
Apply Empty_Subq_eq to the current goal.
Let y be given.
Assume Hy: y ∈ U ∩ (A ∩ B).
We prove the intermediate claim HyU: y ∈ U.
An exact proof term for the current goal is (binintersectE1 U (A ∩ B) y Hy).
We prove the intermediate claim HyAB: y ∈ A ∩ B.
An exact proof term for the current goal is (binintersectE2 U (A ∩ B) y Hy).
We prove the intermediate claim HyA: y ∈ A.
An exact proof term for the current goal is (binintersectE1 A B y HyAB).
We prove the intermediate claim HyUA: y ∈ U ∩ A.
Apply binintersectI to the current goal.
An exact proof term for the current goal is HyU.
An exact proof term for the current goal is HyA.
rewrite the current goal using Hempty (from right to left).
An exact proof term for the current goal is HyUA.
An exact proof term for the current goal is (SepI X (λx0 ⇒ ∀U : set, U ∈ Tx → x0 ∈ U → U ∩ A ≠ Empty) x HxX (andER (x ∈ X) (∀U : set, U ∈ Tx → x ∈ U → U ∩ A ≠ Empty) HxA)).
We will prove x ∈ closure_of X Tx B.
We prove the intermediate claim HxB: x ∈ X ∧ (∀U : set, U ∈ Tx → x ∈ U → U ∩ B ≠ Empty).
Apply andI to the current goal.
An exact proof term for the current goal is HxX.
Let U be given.
Assume HU: U ∈ Tx.
Assume HxU: x ∈ U.
We will prove U ∩ B ≠ Empty.
We prove the intermediate claim HABne: U ∩ (A ∩ B) ≠ Empty.
An exact proof term for the current goal is (HxAB U HU HxU).
Assume Hempty: U ∩ B = Empty.
Apply HABne to the current goal.
Apply Empty_Subq_eq to the current goal.
Let y be given.
Assume Hy: y ∈ U ∩ (A ∩ B).
We prove the intermediate claim HyU: y ∈ U.
An exact proof term for the current goal is (binintersectE1 U (A ∩ B) y Hy).
We prove the intermediate claim HyAB: y ∈ A ∩ B.
An exact proof term for the current goal is (binintersectE2 U (A ∩ B) y Hy).
We prove the intermediate claim HyB: y ∈ B.
An exact proof term for the current goal is (binintersectE2 A B y HyAB).
We prove the intermediate claim HyUB: y ∈ U ∩ B.
Apply binintersectI to the current goal.
An exact proof term for the current goal is HyU.
An exact proof term for the current goal is HyB.
rewrite the current goal using Hempty (from right to left).
An exact proof term for the current goal is HyUB.
An exact proof term for the current goal is (SepI X (λx0 ⇒ ∀U : set, U ∈ Tx → x0 ∈ U → U ∩ B ≠ Empty) x HxX (andER (x ∈ X) (∀U : set, U ∈ Tx → x ∈ U → U ∩ B ≠ Empty) HxB)).
∎
Theorem. (ex17_9_closure_of_product_subset)
∀X Y Tx Ty A B : set, topology_on X Tx → topology_on Y Ty → closure_of (OrderedPair X Y) (product_topology X Tx Y Ty) (OrderedPair A B) = OrderedPair (closure_of X Tx A) (closure_of Y Ty B)
In Proofgold the corresponding term root is 1e639d... and proposition id is 23e3a5...
Proof:
Let X, Y, Tx, Ty, A and B be given.
Assume HTx: topology_on X Tx.
Assume HTy: topology_on Y Ty.
We will prove closure_of (OrderedPair X Y) (product_topology X Tx Y Ty) (OrderedPair A B) = OrderedPair (closure_of X Tx A) (closure_of Y Ty B).
The rest of this subproof is missing.
∎
Theorem. (ex17_10_order_topology_Hausdorff)
∀X : set, Hausdorff_space X (order_topology X)
In Proofgold the corresponding term root is eb3c31... and proposition id is f42cb8...
Proof:
Let X be given.
We will prove Hausdorff_space X (order_topology X).
The rest of this subproof is missing.
∎
Theorem. (ex17_11_product_Hausdorff)
∀X Tx Y Ty : set, Hausdorff_space X Tx → Hausdorff_space Y Ty → Hausdorff_space (OrderedPair X Y) (product_topology X Tx Y Ty)
In Proofgold the corresponding term root is de34ea... and proposition id is a4d56b...
Proof:
Let X, Tx, Y and Ty be given.
Assume HX: Hausdorff_space X Tx.
Assume HY: Hausdorff_space Y Ty.
We will prove Hausdorff_space (OrderedPair X Y) (product_topology X Tx Y Ty).
The rest of this subproof is missing.
∎
Theorem. (ex17_12_subspace_Hausdorff)
∀X Tx Y : set, Hausdorff_space X Tx → Hausdorff_space Y (subspace_topology X Tx Y)
In Proofgold the corresponding term root is b6b742... and proposition id is ca4106...
Proof:
Let X, Tx and Y be given.
Assume HX: Hausdorff_space X Tx.
We will prove Hausdorff_space Y (subspace_topology X Tx Y).
We prove the intermediate claim HTx: topology_on X Tx.
An exact proof term for the current goal is (andEL (topology_on X Tx) (∀x1 x2 : set, x1 ≠ x2 → ∃U V : set, U ∈ Tx ∧ V ∈ Tx ∧ x1 ∈ U ∧ x2 ∈ V ∧ U ∩ V = Empty) HX).
We prove the intermediate claim HYsubX: Y ⊆ X.
The rest of this subproof is missing.
We prove the intermediate claim HTy: topology_on Y (subspace_topology X Tx Y).
An exact proof term for the current goal is (subspace_topology_is_topology X Tx Y HTx HYsubX).
We will prove topology_on Y (subspace_topology X Tx Y) ∧ (∀y1 y2 : set, y1 ≠ y2 → ∃U V : set, U ∈ subspace_topology X Tx Y ∧ V ∈ subspace_topology X Tx Y ∧ y1 ∈ U ∧ y2 ∈ V ∧ U ∩ V = Empty).
Apply andI to the current goal.
An exact proof term for the current goal is HTy.
Let y1 and y2 be given.
Assume Hne: y1 ≠ y2.
We will prove ∃U V : set, U ∈ subspace_topology X Tx Y ∧ V ∈ subspace_topology X Tx Y ∧ y1 ∈ U ∧ y2 ∈ V ∧ U ∩ V = Empty.
The rest of this subproof is missing.
∎
Theorem. (ex17_13_diagonal_closed_iff_Hausdorff)
∀X Tx : set, topology_on X Tx → (Hausdorff_space X Tx ↔ closed_in (OrderedPair X X) (product_topology X Tx X Tx) {OrderedPair x x|x ∈ X})
In Proofgold the corresponding term root is 88e65c... and proposition id is cdecb2...
Proof:
Let X and Tx be given.
Assume Htop: topology_on X Tx.
We will prove Hausdorff_space X Tx ↔ closed_in (OrderedPair X X) (product_topology X Tx X Tx) {OrderedPair x x|x ∈ X}.
The rest of this subproof is missing.
∎
Theorem. (ex17_14_sequence_in_finite_complement_topology)
∀X seq : set, function_on seq ω X → ∀x : set, x ∈ X → ∀U : set, U ∈ finite_complement_topology X → x ∈ U → ∃N : set, N ∈ ω ∧ ∀n : set, n ∈ ω → N ⊆ n → apply_fun seq n ∈ U
In Proofgold the corresponding term root is 9e8778... and proposition id is 836801...
Proof:
Let X and seq be given.
Assume Hseq: function_on seq ω X.
Let x be given.
Assume Hx: x ∈ X.
Let U be given.
Assume HU: U ∈ finite_complement_topology X.
Assume HxU: x ∈ U.
We will prove ∃N : set, N ∈ ω ∧ ∀n : set, n ∈ ω → N ⊆ n → apply_fun seq n ∈ U.
The rest of this subproof is missing.
∎
Theorem. (ex17_15_T1_characterization)
∀X Tx : set, topology_on X Tx → (T1_space X Tx ↔ (∀x : set, closed_in X Tx {x}))
In Proofgold the corresponding term root is e11025... and proposition id is 8d65cb...
Proof:
Let X and Tx be given.
Assume Htop: topology_on X Tx.
We will prove T1_space X Tx ↔ (∀x : set, closed_in X Tx {x}).
The rest of this subproof is missing.
∎
Theorem. (ex17_16_closure_of_K_in_various_topologies)
closure_of R R_standard_topology K_set ≠ closure_of R R_lower_limit_topology K_set
In Proofgold the corresponding term root is f943d3... and proposition id is 87a1cc...
Proof:
We will prove closure_of R R_standard_topology K_set ≠ closure_of R R_lower_limit_topology K_set.
The rest of this subproof is missing.
∎
Theorem. (ex17_17_closures_in_lower_limit_and_C_topology)
closure_of R R_lower_limit_topology K_set ≠ closure_of R R_K_topology K_set
In Proofgold the corresponding term root is e4c04f... and proposition id is b02fd1...
Proof:
We will prove closure_of R R_lower_limit_topology K_set ≠ closure_of R R_K_topology K_set.
The rest of this subproof is missing.
∎
Theorem. (ex17_18_closures_in_ordered_square)
∃A : set, A ⊆ ordered_square ∧ closure_of ordered_square ordered_square_topology A ≠ closure_of ordered_square ordered_square_subspace_topology A
In Proofgold the corresponding term root is 3d3cdf... and proposition id is 0c0c86...
Proof:
We will prove ∃A : set, A ⊆ ordered_square ∧ closure_of ordered_square ordered_square_topology A ≠ closure_of ordered_square ordered_square_subspace_topology A.
The rest of this subproof is missing.
∎
Definition. We define boundary_of to be λX Tx A ⇒ closure_of X Tx A ∩ closure_of X Tx (X ∖ A) of type set → set → set → set.
In Proofgold the corresponding term root is da6db2... and object id is 6a949d...
Theorem. (ex17_19_boundary_properties)
∀X Tx A : set, topology_on X Tx → boundary_of X Tx A ⊆ closure_of X Tx A ∧ boundary_of X Tx A ⊆ closure_of X Tx (X ∖ A)
In Proofgold the corresponding term root is 9cecb1... and proposition id is e14723...
Proof:
Let X, Tx and A be given.
Assume Htop: topology_on X Tx.
We will prove boundary_of X Tx A ⊆ closure_of X Tx A ∧ boundary_of X Tx A ⊆ closure_of X Tx (X ∖ A).
Apply andI to the current goal.
We will prove boundary_of X Tx A ⊆ closure_of X Tx A.
An exact proof term for the current goal is (binintersect_Subq_1 (closure_of X Tx A) (closure_of X Tx (X ∖ A))).
We will prove boundary_of X Tx A ⊆ closure_of X Tx (X ∖ A).
An exact proof term for the current goal is (binintersect_Subq_2 (closure_of X Tx A) (closure_of X Tx (X ∖ A))).
∎
Theorem. (ex17_20_boundaries_and_interiors_in_R2)
boundary_of (OrderedPair R R) R2_standard_topology ordered_square_open_strip ≠ boundary_of (OrderedPair R R) R2_dictionary_order_topology ordered_square_open_strip
In Proofgold the corresponding term root is 5b0878... and proposition id is c1ef9f...
Theorem. (ex17_21_Kuratowski_closure_complement_maximal)
∀X : set, closure_of X (discrete_topology X) (X ∖ Empty) = X
In Proofgold the corresponding term root is 9bcfc3... and proposition id is 87a00f...
Proof:
Let X be given.
We will prove closure_of X (discrete_topology X) (X ∖ Empty) = X.
We prove the intermediate claim Htop: topology_on X (discrete_topology X).
An exact proof term for the current goal is (discrete_topology_on X).
We prove the intermediate claim HXE: X ∖ Empty = X.
Apply set_ext to the current goal.
Let x be given.
Assume Hx: x ∈ X ∖ Empty.
An exact proof term for the current goal is (setminusE1 X Empty x Hx).
Let x be given.
Assume Hx: x ∈ X.
Apply setminusI to the current goal.
An exact proof term for the current goal is Hx.
Assume Hfalse: x ∈ Empty.
An exact proof term for the current goal is (EmptyE x Hfalse).
rewrite the current goal using HXE (from left to right).
Apply set_ext to the current goal.
An exact proof term for the current goal is (closure_in_space X (discrete_topology X) X Htop).
An exact proof term for the current goal is (subset_of_closure X (discrete_topology X) X Htop (Subq_ref X)).
∎
Definition. We define preimage_of to be λX f V ⇒ {x ∈ X|apply_fun f x ∈ V} of type set → set → set → set.
In Proofgold the corresponding term root is 3b22f6... and object id is c5099c...
Definition. We define continuous_map to be λX Tx Y Ty f ⇒ topology_on X Tx ∧ topology_on Y Ty ∧ function_on f X Y ∧ ∀V : set, V ∈ Ty → preimage_of X f V ∈ Tx of type set → set → set → set → set → prop.
In Proofgold the corresponding term root is 9d141b... and object id is 4cd657...
Definition. We define continuous_at to be λf x ⇒ function_on f R R ∧ x ∈ R ∧ ∀eps : set, eps ∈ R → ∃delta : set, delta ∈ R ∧ True of type set → set → prop.
In Proofgold the corresponding term root is 3fae79... and object id is 3c67da...
Theorem. (continuity_equiv_forms)
∀X Tx Y Ty f : set, topology_on X Tx → topology_on Y Ty → (continuous_map X Tx Y Ty f ↔ (∀V : set, V ∈ Ty → preimage_of X f V ∈ Tx) ∧ (∀C : set, closed_in Y Ty C → closed_in X Tx (preimage_of X f C)) ∧ (∀x : set, x ∈ X → ∀V : set, V ∈ Ty → apply_fun f x ∈ V → ∃U : set, U ∈ Tx ∧ x ∈ U ∧ ∀u : set, u ∈ U → apply_fun f u ∈ V))
In Proofgold the corresponding term root is 4f5d86... and proposition id is 4e94f1...
Proof:
Let X, Tx, Y, Ty and f be given.
Assume HTx: topology_on X Tx.
Assume HTy: topology_on Y Ty.
We will prove continuous_map X Tx Y Ty f ↔ (∀V : set, V ∈ Ty → preimage_of X f V ∈ Tx) ∧ (∀C : set, closed_in Y Ty C → closed_in X Tx (preimage_of X f C)) ∧ (∀x : set, x ∈ X → ∀V : set, V ∈ Ty → apply_fun f x ∈ V → ∃U : set, U ∈ Tx ∧ x ∈ U ∧ ∀u : set, u ∈ U → apply_fun f u ∈ V).
The rest of this subproof is missing.
∎
Theorem. (identity_continuous)
∀X Tx : set, topology_on X Tx → let id ≔ {UPair x x|x ∈ X} in continuous_map X Tx X Tx id
In Proofgold the corresponding term root is 237cd7... and proposition id is 0975bd...
Proof:
Let X and Tx be given.
Assume HTx: topology_on X Tx.
We will prove let id ≔ {UPair x x|x ∈ X} in continuous_map X Tx X Tx id.
The rest of this subproof is missing.
∎
Definition. We define compose_fun to be λX f g ⇒ {UPair x (apply_fun g (apply_fun f x))|x ∈ X} of type set → set → set → set.
In Proofgold the corresponding term root is a11861... and object id is cb0506...
Theorem. (composition_continuous)
∀X Tx Y Ty Z Tz f g : set, continuous_map X Tx Y Ty f → continuous_map Y Ty Z Tz g → continuous_map X Tx Z Tz (compose_fun X f g)
In Proofgold the corresponding term root is e16478... and proposition id is 7f3da0...
Proof:
Let X, Tx, Y, Ty, Z, Tz, f and g be given.
Assume Hf: continuous_map X Tx Y Ty f.
Assume Hg: continuous_map Y Ty Z Tz g.
We will prove continuous_map X Tx Z Tz (compose_fun X f g).
The rest of this subproof is missing.
∎
Theorem. (continuous_construction_rules)
∀X Tx Y Ty Z Tz f g : set, continuous_map X Tx Y Ty f → continuous_map X Tx Y Ty g → continuous_map X Tx Y Ty f ∧ continuous_map X Tx Y Ty g ∧ continuous_map X Tx Y Ty g
In Proofgold the corresponding term root is 89b71f... and proposition id is 0c5709...
Proof:
Let X, Tx, Y, Ty, Z, Tz, f and g be given.
Assume Hf: continuous_map X Tx Y Ty f.
Assume Hg: continuous_map X Tx Y Ty g.
We will prove continuous_map X Tx Y Ty f ∧ continuous_map X Tx Y Ty g ∧ continuous_map X Tx Y Ty g.
An exact proof term for the current goal is (andI (continuous_map X Tx Y Ty f ∧ continuous_map X Tx Y Ty g) (continuous_map X Tx Y Ty g) (andI (continuous_map X Tx Y Ty f) (continuous_map X Tx Y Ty g) Hf Hg) Hg).
∎
Definition. We define homeomorphism to be λX Tx Y Ty f ⇒ continuous_map X Tx Y Ty f ∧ ∃g : set, continuous_map Y Ty X Tx g ∧ (∀x : set, x ∈ X → apply_fun g (apply_fun f x) = x) ∧ (∀y : set, y ∈ Y → apply_fun f (apply_fun g y) = y) of type set → set → set → set → set → prop.
In Proofgold the corresponding term root is 6e081c... and object id is 701119...
Theorem. (continuous_on_subspace)
∀X Tx Y Ty f A : set, topology_on X Tx → A ⊆ X → continuous_map X Tx Y Ty f → continuous_map A (subspace_topology X Tx A) Y Ty f
In Proofgold the corresponding term root is 772a7b... and proposition id is cc28cd...
Proof:
Let X, Tx, Y, Ty, f and A be given.
Assume HTx: topology_on X Tx.
Assume HA: A ⊆ X.
Assume Hf: continuous_map X Tx Y Ty f.
We will prove continuous_map A (subspace_topology X Tx A) Y Ty f.
The rest of this subproof is missing.
∎
Theorem. (homeomorphism_inverse_continuous)
∀X Tx Y Ty f : set, homeomorphism X Tx Y Ty f → continuous_map Y Ty X Tx f
In Proofgold the corresponding term root is 0e9573... and proposition id is 0c60bb...
Proof:
Let X, Tx, Y, Ty and f be given.
Assume Hhom: homeomorphism X Tx Y Ty f.
We will prove continuous_map Y Ty X Tx f.
The rest of this subproof is missing.
∎
Theorem. (pasting_lemma)
∀X A B Y Tx Ty f g : set, topology_on X Tx → A ∈ Tx → B ∈ Tx → A ∩ B = Empty → continuous_map A (subspace_topology X Tx A) Y Ty f → continuous_map B (subspace_topology X Tx B) Y Ty g → continuous_map (A ∪ B) (subspace_topology X Tx (A ∪ B)) Y Ty (f ∪ g)
In Proofgold the corresponding term root is 0ae433... and proposition id is 9ef0ff...
Proof:
Let X, A, B, Y, Tx, Ty, f and g be given.
Assume HTx: topology_on X Tx.
Assume HA: A ∈ Tx.
Assume HB: B ∈ Tx.
Assume Hdisj: A ∩ B = Empty.
Assume Hf: continuous_map A (subspace_topology X Tx A) Y Ty f.
Assume Hg: continuous_map B (subspace_topology X Tx B) Y Ty g.
We will prove continuous_map (A ∪ B) (subspace_topology X Tx (A ∪ B)) Y Ty (f ∪ g).
The rest of this subproof is missing.
∎
Theorem. (maps_into_products)
∀A X Tx Y Ty f g : set, continuous_map A Tx X Ty f → continuous_map A Tx Y Ty g → continuous_map A Tx (OrderedPair X Y) (product_topology X Ty Y Ty) (f ∩ g)
In Proofgold the corresponding term root is 9691b7... and proposition id is 908ecc...
Proof:
Let A, X, Tx, Y, Ty, f and g be given.
Assume Hf: continuous_map A Tx X Ty f.
Assume Hg: continuous_map A Tx Y Ty g.
We will prove continuous_map A Tx (OrderedPair X Y) (product_topology X Ty Y Ty) (f ∩ g).
The rest of this subproof is missing.
∎
Definition. We define projection_map to be λX Y ⇒ projection1 X Y of type set → set → set.
In Proofgold the corresponding term root is ca91af... and object id is 8a192b...
Theorem. (projections_are_continuous)
∀X Tx Y Ty : set, topology_on X Tx → topology_on Y Ty → continuous_map (OrderedPair X Y) (product_topology X Tx Y Ty) X Tx (projection_map X Y) ∧ continuous_map (OrderedPair X Y) (product_topology X Tx Y Ty) Y Ty (projection_map Y X)
In Proofgold the corresponding term root is 237b1a... and proposition id is d5993e...
Proof:
Let X, Tx, Y and Ty be given.
Assume HTx: topology_on X Tx.
Assume HTy: topology_on Y Ty.
We will prove continuous_map (OrderedPair X Y) (product_topology X Tx Y Ty) X Tx (projection_map X Y) ∧ continuous_map (OrderedPair X Y) (product_topology X Tx Y Ty) Y Ty (projection_map Y X).
The rest of this subproof is missing.
∎
Theorem. (product_topology_universal)
∀X Tx Y Ty : set, topology_on X Tx → topology_on Y Ty → ∃Tprod : set, topology_on (OrderedPair X Y) Tprod ∧ continuous_map (OrderedPair X Y) Tprod X Tx (projection_map X Y) ∧ continuous_map (OrderedPair X Y) Tprod Y Ty (projection_map Y X)
In Proofgold the corresponding term root is 07f793... and proposition id is 1c46db...
Proof:
Let X, Tx, Y and Ty be given.
Assume HTx: topology_on X Tx.
Assume HTy: topology_on Y Ty.
We will prove ∃Tprod : set, topology_on (OrderedPair X Y) Tprod ∧ continuous_map (OrderedPair X Y) Tprod X Tx (projection_map X Y) ∧ continuous_map (OrderedPair X Y) Tprod Y Ty (projection_map Y X).
The rest of this subproof is missing.
∎
Definition. We define metric_on to be λX d ⇒ function_on d (OrderedPair X X) R ∧ (∀x y : set, x ∈ X → y ∈ X → apply_fun d (OrderedPair x y) = apply_fun d (OrderedPair y x)) ∧ (∀x : set, x ∈ X → apply_fun d (OrderedPair x x) = 0) ∧ (∀x y : set, x ∈ X → y ∈ X → ¬ (Rlt (apply_fun d (OrderedPair x y)) 0) ∧ apply_fun d (OrderedPair x y) = 0 → x = y) ∧ (∀x y z : set, x ∈ X → y ∈ X → z ∈ X → Rlt (apply_fun d (OrderedPair x z)) (apply_fun d (OrderedPair x y) ∩ apply_fun d (OrderedPair y z))) of type set → set → prop.
In Proofgold the corresponding term root is 133c6d... and object id is 44e3e2...
Definition. We define open_ball to be λX d x ⇒ {y ∈ X|∃r ∈ R, Rlt (d x y) r} of type set → set → set → set.
In Proofgold the corresponding term root is edb224... and object id is 982685...
Definition. We define metric_topology to be λX d ⇒ generated_topology X {open_ball X d x|x ∈ X} of type set → set → set.
In Proofgold the corresponding term root is ec09f6... and object id is 1d46e4...
Theorem. (open_balls_form_basis)
∀X d : set, metric_on X d → basis_on X {open_ball X d x|x ∈ X}
In Proofgold the corresponding term root is 02919c... and proposition id is e09577...
Proof:
Let X and d be given.
Assume Hd: metric_on X d.
We will prove basis_on X {open_ball X d x|x ∈ X}.
The rest of this subproof is missing.
∎
Theorem. (metric_topology_is_topology)
∀X d : set, metric_on X d → topology_on X (metric_topology X d)
In Proofgold the corresponding term root is ac1fde... and proposition id is 72ac18...
Proof:
Let X and d be given.
Assume Hd: metric_on X d.
We will prove topology_on X (metric_topology X d).
The rest of this subproof is missing.
∎
Theorem. (metric_topology_generated_by_balls)
∀X d : set, metric_on X d → generated_topology X {open_ball X d x|x ∈ X} = metric_topology X d
In Proofgold the corresponding term root is 35ac46... and proposition id is 7fbd55...
Proof:
Let X and d be given.
Assume Hd: metric_on X d.
We will prove generated_topology X {open_ball X d x|x ∈ X} = metric_topology X d.
The rest of this subproof is missing.
∎
Theorem. (metric_epsilon_delta_continuity)
∀X dX Y dY f : set, metric_on X dX → metric_on Y dY → (continuous_map X (metric_topology X dX) Y (metric_topology Y dY) f ↔ (∀x0 : set, x0 ∈ X → ∀eps : set, eps ∈ R ∧ Rlt 0 eps → ∃delta : set, delta ∈ R ∧ Rlt 0 delta ∧ (∀x : set, x ∈ X → Rlt (apply_fun dX (OrderedPair x x0)) delta → Rlt (apply_fun dY (OrderedPair (apply_fun f x) (apply_fun f x0))) eps)))
In Proofgold the corresponding term root is ec975a... and proposition id is e19b7c...
Proof:
Let X, dX, Y, dY and f be given.
Assume HdX: metric_on X dX.
Assume HdY: metric_on Y dY.
We will prove continuous_map X (metric_topology X dX) Y (metric_topology Y dY) f ↔ (∀x0 : set, x0 ∈ X → ∀eps : set, eps ∈ R ∧ Rlt 0 eps → ∃delta : set, delta ∈ R ∧ Rlt 0 delta ∧ (∀x : set, x ∈ X → Rlt (apply_fun dX (OrderedPair x x0)) delta → Rlt (apply_fun dY (OrderedPair (apply_fun f x) (apply_fun f x0))) eps)).
The rest of this subproof is missing.
∎
Definition. We define sequence_in to be λseq A ⇒ seq ⊆ A of type set → set → prop.
In Proofgold the corresponding term root is 81c0ef... and object id is 317007...
Definition. We define sequence_on to be λseq A ⇒ function_on seq ω A of type set → set → prop.
In Proofgold the corresponding term root is 8a621a... and object id is d8b716...
Definition. We define converges_to to be λX Tx seq x ⇒ topology_on X Tx ∧ sequence_on seq X ∧ x ∈ X ∧ ∀U : set, U ∈ Tx → x ∈ U → ∃N : set, N ∈ ω ∧ ∀n : set, n ∈ ω → N ⊆ n → apply_fun seq n ∈ U of type set → set → set → set → prop.
In Proofgold the corresponding term root is 166c07... and object id is 06f756...
Definition. We define image_of to be λf seq ⇒ Repl seq (λy ⇒ y) of type set → set → set.
In Proofgold the corresponding term root is 129b50... and object id is 2c7a62...
Definition. We define function_sequence_value to be λf_seq n x ⇒ apply_fun (apply_fun f_seq n) x of type set → set → set → set.
In Proofgold the corresponding term root is 375137... and object id is e780e4...
Definition. We define sequence_converges_metric to be λX d seq x ⇒ metric_on X d ∧ sequence_on seq X ∧ x ∈ X ∧ ∀eps : set, eps ∈ R ∧ Rlt 0 eps → ∃N : set, N ∈ ω ∧ ∀n : set, n ∈ ω → N ⊆ n → Rlt (apply_fun d (OrderedPair (apply_fun seq n) x)) eps of type set → set → set → set → prop.
In Proofgold the corresponding term root is 91baf1... and object id is c21c73...
Theorem. (metric_limits_unique)
∀X d seq x y : set, metric_on X d → sequence_on seq X → sequence_converges_metric X d seq x → sequence_converges_metric X d seq y → x = y
In Proofgold the corresponding term root is 4a0052... and proposition id is 9bb316...
Proof:
Let X, d, seq, x and y be given.
Assume Hd: metric_on X d.
Assume Hseq: sequence_on seq X.
Assume Hx: sequence_converges_metric X d seq x.
Assume Hy: sequence_converges_metric X d seq y.
We will prove x = y.
The rest of this subproof is missing.
∎
Definition. We define uniform_convergence_functions to be λX dX Y dY f_seq f ⇒ metric_on X dX ∧ metric_on Y dY ∧ function_on f_seq ω (function_space X Y) ∧ function_on f X Y ∧ (∀n : set, n ∈ ω → function_on (apply_fun f_seq n) X Y) ∧ ∀eps : set, eps ∈ R ∧ Rlt 0 eps → ∃N : set, N ∈ ω ∧ ∀n : set, n ∈ ω → N ⊆ n → ∀x : set, x ∈ X → Rlt (apply_fun dY (OrderedPair (apply_fun (apply_fun f_seq n) x) (apply_fun f x))) eps of type set → set → set → set → set → set → prop.
In Proofgold the corresponding term root is 689b3c... and object id is 45183f...
Theorem. (uniform_limit_of_continuous_is_continuous)
∀X dX Y dY f_seq f : set, metric_on X dX → metric_on Y dY → function_on f_seq ω (function_space X Y) → (∀n : set, n ∈ ω → continuous_map X (metric_topology X dX) Y (metric_topology Y dY) (apply_fun f_seq n)) → uniform_convergence_functions X dX Y dY f_seq f → continuous_map X (metric_topology X dX) Y (metric_topology Y dY) f
In Proofgold the corresponding term root is 6ec8f8... and proposition id is 23f859...
Proof:
Let X, dX, Y, dY, f_seq and f be given.
Assume HdX: metric_on X dX.
Assume HdY: metric_on Y dY.
Assume Hfseq: function_on f_seq ω (function_space X Y).
Assume Hcont: ∀n : set, n ∈ ω → continuous_map X (metric_topology X dX) Y (metric_topology Y dY) (apply_fun f_seq n).
Assume Hunif: uniform_convergence_functions X dX Y dY f_seq f.
We will prove continuous_map X (metric_topology X dX) Y (metric_topology Y dY) f.
The rest of this subproof is missing.
∎
Theorem. (sequence_convergence_metric)
∀X d seq x : set, sequence_converges_metric X d seq x → sequence_converges_metric X d seq x
In Proofgold the corresponding term root is 94e491... and proposition id is 50e1d5...
Proof:
Let X, d, seq and x be given.
Assume H: sequence_converges_metric X d seq x.
We will prove sequence_converges_metric X d seq x.
An exact proof term for the current goal is H.
∎
Theorem. (continuity_via_sequences_metric)
∀X dX Y dY f : set, metric_on X dX → metric_on Y dY → (continuous_map X (metric_topology X dX) Y (metric_topology Y dY) f ↔ ∀seq x : set, sequence_converges_metric X dX seq x → sequence_converges_metric Y dY ({OrderedPair n (apply_fun f (apply_fun seq n))|n ∈ ω}) (apply_fun f x))
In Proofgold the corresponding term root is 061987... and proposition id is 522f60...
Proof:
Let X, dX, Y, dY and f be given.
Assume HdX: metric_on X dX.
Assume HdY: metric_on Y dY.
We will prove continuous_map X (metric_topology X dX) Y (metric_topology Y dY) f ↔ ∀seq x : set, sequence_converges_metric X dX seq x → sequence_converges_metric Y dY ({OrderedPair n (apply_fun f (apply_fun seq n))|n ∈ ω}) (apply_fun f x).
The rest of this subproof is missing.
∎
Definition. We define quotient_topology to be λX Tx Y f ⇒ {V ∈ 𝒫 Y|{x ∈ X|apply_fun f x ∈ V} ∈ Tx} of type set → set → set → set → set.
In Proofgold the corresponding term root is c0aec3... and object id is 7298c4...
Definition. We define quotient_map to be λX Tx Y f ⇒ topology_on X Tx ∧ function_on f X Y ∧ (∀y : set, y ∈ Y → ∃x : set, x ∈ X ∧ apply_fun f x = y) of type set → set → set → set → prop.
In Proofgold the corresponding term root is 3c7f20... and object id is ad7056...
Theorem. (quotient_topology_is_topology)
∀X Tx Y f : set, topology_on X Tx → quotient_map X Tx Y f → topology_on Y (quotient_topology X Tx Y f)
In Proofgold the corresponding term root is 8fa568... and proposition id is 167fa2...
Proof:
Let X, Tx, Y and f be given.
Assume HTx: topology_on X Tx.
Assume Hf: quotient_map X Tx Y f.
We will prove topology_on Y (quotient_topology X Tx Y f).
The rest of this subproof is missing.
∎
Theorem. (quotient_universal_property)
∀X Tx Y Ty f : set, quotient_map X Tx Y f → topology_on Y Ty → continuous_map X Tx Y Ty f
In Proofgold the corresponding term root is df1c99... and proposition id is 9ee3ff...
Proof:
Let X, Tx, Y, Ty and f be given.
Assume Hf: quotient_map X Tx Y f.
Assume HTy: topology_on Y Ty.
We will prove continuous_map X Tx Y Ty f.
The rest of this subproof is missing.
∎
Definition. We define separation_of to be λX U V ⇒ U ∈ 𝒫 X ∧ V ∈ 𝒫 X ∧ U ∩ V = Empty ∧ U ≠ Empty ∧ V ≠ Empty of type set → set → set → prop.
In Proofgold the corresponding term root is 52ab80... and object id is 74d9b5...
Definition. We define connected_space to be λX Tx ⇒ topology_on X Tx ∧ ¬ (∃U V : set, U ∈ Tx ∧ V ∈ Tx ∧ separation_of X U V ∧ U ∪ V = X) of type set → set → prop.
In Proofgold the corresponding term root is 8c317d... and object id is 9db60d...
Theorem. (connected_iff_no_nontrivial_clopen)
∀X Tx : set, connected_space X Tx ↔ ¬ (∃A : set, A ≠ Empty ∧ A ≠ X ∧ open_in X Tx A ∧ closed_in X Tx A)
In Proofgold the corresponding term root is d2bef1... and proposition id is 2de3f8...
Proof:
Let X and Tx be given.
We will prove connected_space X Tx ↔ ¬ (∃A : set, A ≠ Empty ∧ A ≠ X ∧ open_in X Tx A ∧ closed_in X Tx A).
The rest of this subproof is missing.
∎
Theorem. (separation_subspace_limit_points)
∀X Tx Y A B : set, topology_on X Tx → A ∩ B = Empty → A ∪ B = Y → open_in X Tx A → open_in X Tx B → ∃a b : set, limit_point_of X Tx A a ∧ limit_point_of X Tx B b ∧ a ∈ Y ∧ b ∈ Y
In Proofgold the corresponding term root is 15a83a... and proposition id is 8b13ad...
Proof:
Let X, Tx, Y, A and B be given.
Assume HTx: topology_on X Tx.
Assume Hdisj: A ∩ B = Empty.
Assume Hunion: A ∪ B = Y.
Assume HA: open_in X Tx A.
Assume HB: open_in X Tx B.
We will prove ∃a b : set, limit_point_of X Tx A a ∧ limit_point_of X Tx B b ∧ a ∈ Y ∧ b ∈ Y.
The rest of this subproof is missing.
∎
Theorem. (connected_subset_in_separation_side)
∀X Tx C D Y : set, topology_on X Tx → connected_space Y Tx → C ∩ D = Empty → C ∪ D = X → open_in X Tx C → open_in X Tx D → Y ⊆ C ∨ Y ⊆ D
In Proofgold the corresponding term root is 4b3c23... and proposition id is 430ce4...
Proof:
Let X, Tx, C, D and Y be given.
Assume HTx: topology_on X Tx.
Assume HY: connected_space Y Tx.
Assume Hdisj: C ∩ D = Empty.
Assume Hunion: C ∪ D = X.
Assume HC: open_in X Tx C.
Assume HD: open_in X Tx D.
We will prove Y ⊆ C ∨ Y ⊆ D.
The rest of this subproof is missing.
∎
Theorem. (union_connected_common_point)
∀X Tx F : set, topology_on X Tx → (∀C : set, C ∈ F → connected_space C (subspace_topology X Tx C)) → (∃x : set, ∀C : set, C ∈ F → x ∈ C) → connected_space (⋃ F) (subspace_topology X Tx (⋃ F))
In Proofgold the corresponding term root is aed708... and proposition id is 2140ac...
Proof:
Let X, Tx and F be given.
Assume HTx: topology_on X Tx.
Assume HF: ∀C : set, C ∈ F → connected_space C (subspace_topology X Tx C).
Assume Hcommon: ∃x : set, ∀C : set, C ∈ F → x ∈ C.
We will prove connected_space (⋃ F) (subspace_topology X Tx (⋃ F)).
The rest of this subproof is missing.
∎
Theorem. (connected_with_limit_points)
∀X Tx A b : set, topology_on X Tx → connected_space A (subspace_topology X Tx A) → limit_point_of X Tx A b → connected_space (A ∪ {b}) (subspace_topology X Tx (A ∪ {b}))
In Proofgold the corresponding term root is 9d9795... and proposition id is 3b6a8f...
Proof:
Let X, Tx, A and b be given.
Assume HTx: topology_on X Tx.
Assume HA: connected_space A (subspace_topology X Tx A).
Assume Hb: limit_point_of X Tx A b.
We will prove connected_space (A ∪ {b}) (subspace_topology X Tx (A ∪ {b})).
The rest of this subproof is missing.
∎
Theorem. (continuous_image_connected)
∀X Tx Y Ty f : set, connected_space X Tx → continuous_map X Tx Y Ty f → connected_space Y Ty
In Proofgold the corresponding term root is b064bb... and proposition id is 866f1a...
Proof:
Let X, Tx, Y, Ty and f be given.
Assume HX: connected_space X Tx.
Assume Hf: continuous_map X Tx Y Ty f.
We will prove connected_space Y Ty.
The rest of this subproof is missing.
∎
Theorem. (interval_connected)
connected_space R R_standard_topology
In Proofgold the corresponding term root is 0bb763... and proposition id is 27f2a5...
Proof:
We will prove connected_space R R_standard_topology.
The rest of this subproof is missing.
∎
Theorem. (connected_subsets_real_are_intervals)
∀A : set, A ⊆ R → connected_space A (subspace_topology R R_standard_topology A) → ∀x y z : set, x ∈ A → y ∈ A → z ∈ R → (Rlt x z ∧ Rlt z y ∨ Rlt y z ∧ Rlt z x) → z ∈ A
In Proofgold the corresponding term root is 72d50e... and proposition id is 2098e2...
Proof:
Let A be given.
Assume HA: A ⊆ R.
Assume Hconn: connected_space A (subspace_topology R R_standard_topology A).
Let x, y and z be given.
Assume Hx: x ∈ A.
Assume Hy: y ∈ A.
Assume Hz: z ∈ R.
Assume Hbetw: Rlt x z ∧ Rlt z y ∨ Rlt y z ∧ Rlt z x.
We will prove z ∈ A.
The rest of this subproof is missing.
∎
Theorem. (finite_product_connected)
∀X Tx Y Ty : set, connected_space X Tx → connected_space Y Ty → connected_space (OrderedPair X Y) (product_topology X Tx Y Ty)
In Proofgold the corresponding term root is 26f0c8... and proposition id is 11bd96...
Proof:
Let X, Tx, Y and Ty be given.
Assume HX: connected_space X Tx.
Assume HY: connected_space Y Ty.
We will prove connected_space (OrderedPair X Y) (product_topology X Tx Y Ty).
The rest of this subproof is missing.
∎
Theorem. (R_omega_product_connected)
connected_space (product_space ω (const_family ω R)) (product_topology_full ω (const_family ω R))
In Proofgold the corresponding term root is 009ffb... and proposition id is 50e166...
Proof:
We will prove connected_space (product_space ω (const_family ω R)) (product_topology_full ω (const_family ω R)).
The rest of this subproof is missing.
∎
Theorem. (R_omega_box_not_connected)
¬ connected_space (product_space ω (const_family ω R)) (box_topology ω (const_family ω R))
In Proofgold the corresponding term root is 754419... and proposition id is b17df9...
Proof:
We will prove ¬ connected_space (product_space ω (const_family ω R)) (box_topology ω (const_family ω R)).
The rest of this subproof is missing.
∎
Definition. We define path_between to be λX x y p ⇒ function_on p unit_interval X ∧ apply_fun p 0 = x ∧ apply_fun p 1 = y of type set → set → set → set → prop.
In Proofgold the corresponding term root is a96d2e... and object id is 9c4ef8...
Definition. We define path_connected_space to be λX Tx ⇒ topology_on X Tx ∧ ∀x y : set, x ∈ X → y ∈ X → ∃p : set, path_between X x y p of type set → set → prop.
In Proofgold the corresponding term root is 03fb51... and object id is 1503f6...
Theorem. (path_connected_implies_connected)
∀X Tx : set, path_connected_space X Tx → connected_space X Tx
In Proofgold the corresponding term root is 850753... and proposition id is a55bc2...
Proof:
Let X and Tx be given.
Assume Hpath: path_connected_space X Tx.
We will prove connected_space X Tx.
The rest of this subproof is missing.
∎
Theorem. (punctured_space_path_connected)
path_connected_space (EuclidPlane ∖ {OrderedPair 0 0}) (subspace_topology EuclidPlane R2_standard_topology (EuclidPlane ∖ {OrderedPair 0 0}))
In Proofgold the corresponding term root is a14d32... and proposition id is b4eb6e...
Theorem. (continuous_image_path_connected)
∀X Tx Y Ty f : set, path_connected_space X Tx → continuous_map X Tx Y Ty f → path_connected_space Y Ty
In Proofgold the corresponding term root is ff97d3... and proposition id is e8a224...
Proof:
Let X, Tx, Y, Ty and f be given.
Assume Hpath: path_connected_space X Tx.
Assume Hf: continuous_map X Tx Y Ty f.
We will prove path_connected_space Y Ty.
The rest of this subproof is missing.
∎
Definition. We define path_component_of to be λX Tx x ⇒ {y ∈ X|∃p : set, function_on p unit_interval X ∧ continuous_map unit_interval R_standard_topology X Tx p ∧ apply_fun p 0 = x ∧ apply_fun p 1 = y} of type set → set → set → set.
In Proofgold the corresponding term root is e1e6af... and object id is ff5ee6...
Theorem. (path_components_equivalence_relation)
∀X Tx : set, topology_on X Tx → (∀x : set, x ∈ X → x ∈ path_component_of X Tx x) ∧ (∀x y : set, x ∈ X → y ∈ X → y ∈ path_component_of X Tx x → x ∈ path_component_of X Tx y) ∧ (∀x y z : set, x ∈ X → y ∈ X → z ∈ X → y ∈ path_component_of X Tx x → z ∈ path_component_of X Tx y → z ∈ path_component_of X Tx x)
In Proofgold the corresponding term root is 664bc5... and proposition id is f7165a...
Proof:
Let X and Tx be given.
Assume HTx: topology_on X Tx.
We will prove (∀x : set, x ∈ X → x ∈ path_component_of X Tx x) ∧ (∀x y : set, x ∈ X → y ∈ X → y ∈ path_component_of X Tx x → x ∈ path_component_of X Tx y) ∧ (∀x y z : set, x ∈ X → y ∈ X → z ∈ X → y ∈ path_component_of X Tx x → z ∈ path_component_of X Tx y → z ∈ path_component_of X Tx x).
The rest of this subproof is missing.
∎
Definition. We define component_of to be λX Tx x ⇒ {y ∈ X|∃C : set, connected_space C (subspace_topology X Tx C) ∧ x ∈ C ∧ y ∈ C} of type set → set → set → set.
In Proofgold the corresponding term root is 9df44b... and object id is 305f50...
Definition. We define locally_connected to be λX Tx ⇒ topology_on X Tx ∧ ∀x : set, x ∈ X → ∀U : set, U ∈ Tx → x ∈ U → ∃V : set, V ∈ Tx ∧ x ∈ V ∧ V ⊆ U ∧ connected_space V (subspace_topology X Tx V) of type set → set → prop.
In Proofgold the corresponding term root is 5bc15b... and object id is 8c5ca2...
Definition. We define locally_path_connected to be λX Tx ⇒ topology_on X Tx ∧ ∀x : set, x ∈ X → ∃U : set, U ∈ Tx ∧ x ∈ U ∧ path_connected_space U (subspace_topology X Tx U) of type set → set → prop.
In Proofgold the corresponding term root is 46f496... and object id is 5f7252...
Definition. We define pairwise_disjoint to be λFam ⇒ ∀U V : set, U ∈ Fam → V ∈ Fam → U ≠ V → U ∩ V = Empty of type set → prop.
In Proofgold the corresponding term root is e9cebe... and object id is 27c86b...
Definition. We define covers to be λX U ⇒ ∀x : set, x ∈ X → ∃u : set, u ∈ U ∧ x ∈ u of type set → set → prop.
In Proofgold the corresponding term root is 26937a... and object id is df2459...
Theorem. (path_components_open)
∀X Tx : set, locally_path_connected X Tx → ∀x : set, x ∈ X → open_in X Tx (path_component_of X Tx x)
In Proofgold the corresponding term root is cca6dc... and proposition id is 9237aa...
Proof:
Let X and Tx be given.
Assume Hlpc: locally_path_connected X Tx.
Let x be given.
Assume Hx: x ∈ X.
We will prove open_in X Tx (path_component_of X Tx x).
The rest of this subproof is missing.
∎
Theorem. (components_equal_path_components)
∀X Tx : set, locally_path_connected X Tx → ∀x : set, x ∈ X → path_component_of X Tx x = component_of X Tx x
In Proofgold the corresponding term root is 577044... and proposition id is a2afa5...
Proof:
Let X and Tx be given.
Assume Hlpc: locally_path_connected X Tx.
Let x be given.
Assume Hx: x ∈ X.
We will prove path_component_of X Tx x = component_of X Tx x.
The rest of this subproof is missing.
∎
Theorem. (components_are_closed)
∀X Tx : set, topology_on X Tx → ∀x : set, x ∈ X → closed_in X Tx (component_of X Tx x)
In Proofgold the corresponding term root is 9b92bd... and proposition id is 7554a6...
Proof:
Let X and Tx be given.
Assume HTx: topology_on X Tx.
Let x be given.
Assume Hx: x ∈ X.
We will prove closed_in X Tx (component_of X Tx x).
The rest of this subproof is missing.
∎
Theorem. (components_partition_space)
∀X Tx : set, topology_on X Tx → covers X {component_of X Tx x|x ∈ X} ∧ pairwise_disjoint {component_of X Tx x|x ∈ X}
In Proofgold the corresponding term root is 3fd51d... and proposition id is a1f8b9...
Proof:
Let X and Tx be given.
Assume HTx: topology_on X Tx.
We will prove covers X {component_of X Tx x|x ∈ X} ∧ pairwise_disjoint {component_of X Tx x|x ∈ X}.
The rest of this subproof is missing.
∎
Theorem. (quotient_preserves_local_connectedness)
∀X Tx Y f : set, quotient_map X Tx Y f → locally_connected X Tx → locally_connected Y (quotient_topology X Tx Y f)
In Proofgold the corresponding term root is 9ddc73... and proposition id is ee7a17...
Proof:
Let X, Tx, Y and f be given.
Assume Hquot: quotient_map X Tx Y f.
Assume Hloc: locally_connected X Tx.
We will prove locally_connected Y (quotient_topology X Tx Y f).
The rest of this subproof is missing.
∎
Definition. We define quasicomponent_of to be λX Tx x ⇒ {y ∈ X|∀U : set, open_in X Tx U → closed_in X Tx U → x ∈ U → y ∈ U} of type set → set → set → set.
In Proofgold the corresponding term root is b1bab5... and object id is 38153d...
Theorem. (components_vs_quasicomponents)
∀X Tx : set, topology_on X Tx → (∀x : set, component_of X Tx x ⊆ quasicomponent_of X Tx x) ∧ (locally_connected X Tx → ∀x : set, component_of X Tx x = quasicomponent_of X Tx x)
In Proofgold the corresponding term root is 57f961... and proposition id is 15c98b...
Proof:
Let X and Tx be given.
Assume HTx: topology_on X Tx.
We will prove (∀x : set, component_of X Tx x ⊆ quasicomponent_of X Tx x) ∧ (locally_connected X Tx → ∀x : set, component_of X Tx x = quasicomponent_of X Tx x).
The rest of this subproof is missing.
∎
Theorem. (ex23_Rl_components)
component_of R R_lower_limit_topology 0 = {0} ∧ (∀x : set, x ∈ R → component_of R R_lower_limit_topology x = {x})
In Proofgold the corresponding term root is 63e30f... and proposition id is fa4e65...
Proof:
We will prove component_of R R_lower_limit_topology 0 = {0} ∧ (∀x : set, x ∈ R → component_of R R_lower_limit_topology x = {x}).
The rest of this subproof is missing.
∎
Theorem. (ex23_Romega_components)
component_of (product_space ω (const_family ω R)) (product_topology_full ω (const_family ω R)) (const_family ω 0) = product_space ω (const_family ω R) ∧ component_of (product_space ω (const_family ω R)) (box_topology ω (const_family ω R)) (const_family ω 0) = {f ∈ product_space ω (const_family ω R)|∃F : set, finite F ∧ ∀i : set, i ∈ ω ∖ F → apply_fun f i = 0}
In Proofgold the corresponding term root is 4e97e0... and proposition id is b88467...
Theorem. (ex23_ordered_square_locally_conn_not_pathconn)
locally_connected ordered_square ordered_square_topology ∧ ¬ locally_path_connected ordered_square ordered_square_topology
In Proofgold the corresponding term root is 025b8e... and proposition id is 084f50...
Theorem. (ex23_connected_open_sets_path_connected)
∀X Tx U : set, locally_path_connected X Tx → open_in X Tx U → connected_space U (subspace_topology X Tx U) → path_connected_space U (subspace_topology X Tx U)
In Proofgold the corresponding term root is 16047c... and proposition id is a3c9df...
Proof:
Let X, Tx and U be given.
Assume Hlpc: locally_path_connected X Tx.
Assume HU: open_in X Tx U.
Assume Hconn: connected_space U (subspace_topology X Tx U).
We will prove path_connected_space U (subspace_topology X Tx U).
The rest of this subproof is missing.
∎
Theorem. (ex23_path_connected_not_locally_connected_examples)
∃A : set, A ⊆ EuclidPlane ∧ path_connected_space A (subspace_topology EuclidPlane R2_standard_topology A) ∧ ¬ locally_connected A (subspace_topology EuclidPlane R2_standard_topology A)
In Proofgold the corresponding term root is 751031... and proposition id is 8fc9b6...
Definition. We define open_cover_of to be λX Tx Fam ⇒ topology_on X Tx ∧ Fam ⊆ 𝒫 X ∧ X ⊆ ⋃ Fam ∧ (∀U : set, U ∈ Fam → U ∈ Tx) of type set → set → set → prop.
In Proofgold the corresponding term root is b602bc... and object id is 45c1d0...
Definition. We define has_finite_subcover to be λX Tx Fam ⇒ ∃G : set, G ⊆ Fam ∧ finite G ∧ X ⊆ ⋃ G of type set → set → set → prop.
In Proofgold the corresponding term root is 2ffe06... and object id is 54d485...
Definition. We define compact_space to be λX Tx ⇒ topology_on X Tx ∧ ∀Fam : set, open_cover_of X Tx Fam → has_finite_subcover X Tx Fam of type set → set → prop.
In Proofgold the corresponding term root is c47023... and object id is 6c5f9a...
Theorem. (Heine_Borel_subcover)
∀X Tx Fam : set, compact_space X Tx → open_cover_of X Tx Fam → has_finite_subcover X Tx Fam
In Proofgold the corresponding term root is 5a6a6b... and proposition id is eb96fe...
Proof:
Let X, Tx and Fam be given.
Assume Hcomp: compact_space X Tx.
Assume HFam: open_cover_of X Tx Fam.
We will prove has_finite_subcover X Tx Fam.
The rest of this subproof is missing.
∎
Theorem. (compact_subspace_via_ambient_covers)
∀X Tx Y : set, topology_on X Tx → (compact_space Y (subspace_topology X Tx Y) ↔ ∀Fam : set, open_cover_of Y Tx Fam → has_finite_subcover Y Tx Fam)
In Proofgold the corresponding term root is 6589b5... and proposition id is b9de63...
Proof:
Let X, Tx and Y be given.
Assume HTx: topology_on X Tx.
We will prove compact_space Y (subspace_topology X Tx Y) ↔ ∀Fam : set, open_cover_of Y Tx Fam → has_finite_subcover Y Tx Fam.
The rest of this subproof is missing.
∎
Theorem. (closed_subspace_compact)
∀X Tx Y : set, compact_space X Tx → closed_in X Tx Y → compact_space Y (subspace_topology X Tx Y)
In Proofgold the corresponding term root is 2f4cec... and proposition id is 49e639...
Proof:
Let X, Tx and Y be given.
Assume Hcomp: compact_space X Tx.
Assume HY: closed_in X Tx Y.
We will prove compact_space Y (subspace_topology X Tx Y).
The rest of this subproof is missing.
∎
Theorem. (compact_subspace_in_Hausdorff_closed)
∀X Tx Y : set, Hausdorff_space X Tx → compact_space Y (subspace_topology X Tx Y) → closed_in X Tx Y
In Proofgold the corresponding term root is 3450b8... and proposition id is babb6c...
Proof:
Let X, Tx and Y be given.
Assume HH: Hausdorff_space X Tx.
Assume Hcomp: compact_space Y (subspace_topology X Tx Y).
We will prove closed_in X Tx Y.
The rest of this subproof is missing.
∎
Theorem. (Hausdorff_separate_point_compact_set)
∀X Tx Y x : set, Hausdorff_space X Tx → compact_space Y (subspace_topology X Tx Y) → x ∩ Y = Empty → ∃U V : set, U ∈ Tx ∧ V ∈ Tx ∧ x ∈ U ∧ Y ⊆ V ∧ U ∩ V = Empty
In Proofgold the corresponding term root is 483b85... and proposition id is 8da4c4...
Proof:
Let X, Tx, Y and x be given.
Assume HH: Hausdorff_space X Tx.
Assume Hcomp: compact_space Y (subspace_topology X Tx Y).
Assume Hx: x ∩ Y = Empty.
We will prove ∃U V : set, U ∈ Tx ∧ V ∈ Tx ∧ x ∈ U ∧ Y ⊆ V ∧ U ∩ V = Empty.
The rest of this subproof is missing.
∎
Theorem. (continuous_image_compact)
∀X Tx Y Ty f : set, compact_space X Tx → continuous_map X Tx Y Ty f → compact_space Y Ty
In Proofgold the corresponding term root is d14440... and proposition id is e0e7d3...
Proof:
Let X, Tx, Y, Ty and f be given.
Assume Hcomp: compact_space X Tx.
Assume Hf: continuous_map X Tx Y Ty f.
We will prove compact_space Y Ty.
The rest of this subproof is missing.
∎
Theorem. (tube_lemma)
∀X Tx Y Ty : set, topology_on X Tx → topology_on Y Ty → compact_space X Tx → ∀x0 : set, x0 ∈ X → ∀N : set, N ∈ product_topology X Tx Y Ty ∧ x0 ∈ N → ∃U : set, U ∈ Tx ∧ x0 ∈ U ∧ (∀y : set, y ∈ Y → OrderedPair U y ∈ N)
In Proofgold the corresponding term root is 9e6d0f... and proposition id is 28302c...
Proof:
Let X, Tx, Y and Ty be given.
Assume HTx: topology_on X Tx.
Assume HTy: topology_on Y Ty.
Assume Hcomp: compact_space X Tx.
Let x0 be given.
Assume Hx0: x0 ∈ X.
Let N be given.
Assume HN: N ∈ product_topology X Tx Y Ty ∧ x0 ∈ N.
We will prove ∃U : set, U ∈ Tx ∧ x0 ∈ U ∧ (∀y : set, y ∈ Y → OrderedPair U y ∈ N).
The rest of this subproof is missing.
∎
Definition. We define bijection to be λX Y f ⇒ function_on f X Y ∧ (∀y : set, y ∈ Y → ∃x : set, x ∈ X ∧ apply_fun f x = y ∧ (∀x' : set, x' ∈ X → apply_fun f x' = y → x' = x)) of type set → set → set → prop.
In Proofgold the corresponding term root is 6c449a... and object id is 5a85b2...
Definition. We define Abs to be abs_SNo of type set → set.
In Proofgold the corresponding term root is 34f6dc... and object id is 2990e5...
Theorem. (compact_to_Hausdorff_bijection_homeomorphism)
∀X Tx Y Ty f : set, compact_space X Tx → Hausdorff_space Y Ty → continuous_map X Tx Y Ty f → bijection X Y f → homeomorphism X Tx Y Ty f
In Proofgold the corresponding term root is dd1587... and proposition id is b5402d...
Proof:
Let X, Tx, Y, Ty and f be given.
Assume Hcomp: compact_space X Tx.
Assume HH: Hausdorff_space Y Ty.
Assume Hcont: continuous_map X Tx Y Ty f.
Assume Hbij: bijection X Y f.
We will prove homeomorphism X Tx Y Ty f.
The rest of this subproof is missing.
∎
Definition. We define bounded_subset_of_reals to be λA ⇒ ∃M : set, M ∈ R ∧ ∀x : set, x ∈ A → ¬ (Rlt M (Abs x)) of type set → prop.
In Proofgold the corresponding term root is fe3ccf... and object id is 6918ab...
Theorem. (finite_product_compact)
∀X Tx Y Ty : set, compact_space X Tx → compact_space Y Ty → compact_space (OrderedPair X Y) (product_topology X Tx Y Ty)
In Proofgold the corresponding term root is 5e7f90... and proposition id is f5be4c...
Proof:
Let X, Tx, Y and Ty be given.
Assume HX: compact_space X Tx.
Assume HY: compact_space Y Ty.
We will prove compact_space (OrderedPair X Y) (product_topology X Tx Y Ty).
The rest of this subproof is missing.
∎
Theorem. (ex26_compactness_exercises)
∀X Tx : set, compact_space X Tx → (closed_in R R_standard_topology unit_interval) ∧ (compact_space unit_interval R_standard_topology)
In Proofgold the corresponding term root is 88f378... and proposition id is 30bf40...
Proof:
Let X and Tx be given.
Assume HX: compact_space X Tx.
We will prove closed_in R R_standard_topology unit_interval ∧ compact_space unit_interval R_standard_topology.
The rest of this subproof is missing.
∎
Theorem. (Heine_Borel_closed_bounded)
∀A : set, A ⊆ R → compact_space A (subspace_topology R R_standard_topology A) → closed_in R R_standard_topology A ∧ bounded_subset_of_reals A
In Proofgold the corresponding term root is d79277... and proposition id is cf79f5...
Proof:
Let A be given.
Assume HA: A ⊆ R.
Assume Hcomp: compact_space A (subspace_topology R R_standard_topology A).
We will prove closed_in R R_standard_topology A ∧ bounded_subset_of_reals A.
The rest of this subproof is missing.
∎
Theorem. (compact_real_closed_bounded)
∀A : set, compact_space A (subspace_topology R R_standard_topology A) → closed_in R R_standard_topology A ∧ bounded_subset_of_reals A
In Proofgold the corresponding term root is 38b967... and proposition id is 969285...
Proof:
Let A be given.
Assume Hcomp: compact_space A (subspace_topology R R_standard_topology A).
We will prove closed_in R R_standard_topology A ∧ bounded_subset_of_reals A.
The rest of this subproof is missing.
∎
Definition. We define limit_point_compact to be λX Tx ⇒ topology_on X Tx ∧ ∀A : set, A ⊆ X → infinite A → ∃x : set, limit_point_of X Tx A x of type set → set → prop.
In Proofgold the corresponding term root is 8e719a... and object id is bc5bd0...
Theorem. (compact_implies_limit_point_compact)
∀X Tx : set, compact_space X Tx → limit_point_compact X Tx
In Proofgold the corresponding term root is f5506e... and proposition id is 34f6b6...
Proof:
Let X and Tx be given.
Assume Hcomp: compact_space X Tx.
We will prove limit_point_compact X Tx.
The rest of this subproof is missing.
∎
Theorem. (limit_point_compact_not_necessarily_compact)
∃X Tx : set, limit_point_compact X Tx ∧ ¬ compact_space X Tx
In Proofgold the corresponding term root is 464ab7... and proposition id is 51bc23...
Proof:
We will prove ∃X Tx : set, limit_point_compact X Tx ∧ ¬ compact_space X Tx.
The rest of this subproof is missing.
∎
Definition. We define locally_compact to be λX Tx ⇒ topology_on X Tx ∧ ∀x : set, x ∈ X → ∃U : set, U ∈ Tx ∧ x ∈ U ∧ compact_space (closure_of X Tx U) (subspace_topology X Tx (closure_of X Tx U)) of type set → set → prop.
In Proofgold the corresponding term root is 47e875... and object id is b40691...
Theorem. (Hausdorff_compact_sets_closed)
∀X Tx A : set, Hausdorff_space X Tx → compact_space A (subspace_topology X Tx A) → closed_in X Tx A
In Proofgold the corresponding term root is 3450b8... and proposition id is babb6c...
Proof:
Let X, Tx and A be given.
Assume HH: Hausdorff_space X Tx.
Assume Hcomp: compact_space A (subspace_topology X Tx A).
We will prove closed_in X Tx A.
The rest of this subproof is missing.
∎
Definition. We define one_point_compactification to be λX Tx Y Ty ⇒ compact_space Y Ty ∧ Hausdorff_space Y Ty ∧ X ⊆ Y ∧ ∃p : set, p ∈ Y ∧ ¬ p ∈ X ∧ subspace_topology Y Ty X = Tx ∧ (∀y : set, y ∈ Y → y ∈ X ∨ y = p) of type set → set → set → set → prop.
In Proofgold the corresponding term root is 76b094... and object id is 90d858...
Theorem. (one_point_compactification_exists)
∀X Tx : set, locally_compact X Tx → Hausdorff_space X Tx → ∃Y Ty : set, one_point_compactification X Tx Y Ty
In Proofgold the corresponding term root is cec952... and proposition id is 92b551...
Proof:
Let X and Tx be given.
Assume Hlc: locally_compact X Tx.
Assume HH: Hausdorff_space X Tx.
We will prove ∃Y Ty : set, one_point_compactification X Tx Y Ty.
The rest of this subproof is missing.
∎
Theorem. (ex29_local_compactness_exercises)
∀X Tx : set, locally_compact X Tx → Hausdorff_space X Tx → ∃Y Ty : set, one_point_compactification X Tx Y Ty
In Proofgold the corresponding term root is cec952... and proposition id is 92b551...
Proof:
Let X and Tx be given.
Assume Hlc: locally_compact X Tx.
Assume HH: Hausdorff_space X Tx.
We will prove ∃Y Ty : set, one_point_compactification X Tx Y Ty.
The rest of this subproof is missing.
∎
Definition. We define directed_set to be λJ ⇒ J ≠ Empty ∧ ∀i j : set, i ∈ J → j ∈ J → ∃k : set, k ∈ J of type set → prop.
In Proofgold the corresponding term root is a64560... and object id is 12321f...
Theorem. (examples_of_directed_sets)
∀J : set, directed_set J → directed_set J
In Proofgold the corresponding term root is 58cbcc... and proposition id is d77c5a...
Proof:
Let J be given.
Assume H: directed_set J.
We will prove directed_set J.
An exact proof term for the current goal is H.
∎
Theorem. (cofinal_subset_directed)
∀J K : set, directed_set J → K ⊆ J → (∀i : set, i ∈ J → ∃k : set, k ∈ K ∧ i ∈ K ∨ i ∈ J) → directed_set K
In Proofgold the corresponding term root is cf3d3a... and proposition id is 1c8417...
Proof:
Let J and K be given.
Assume HJ: directed_set J.
Assume HK: K ⊆ J.
Assume Hcofinal: ∀i : set, i ∈ J → ∃k : set, k ∈ K ∧ i ∈ K ∨ i ∈ J.
We will prove directed_set K.
The rest of this subproof is missing.
∎
Definition. We define net_on to be λnet ⇒ ∃J X : set, directed_set J ∧ function_on net J X of type set → prop.
In Proofgold the corresponding term root is fa568a... and object id is f7a47d...
Definition. We define subnet_of to be λnet sub ⇒ ∃J X K Y phi : set, directed_set J ∧ function_on net J X ∧ directed_set K ∧ function_on sub K Y ∧ function_on phi K J ∧ (∀k1 k2 : set, k1 ∈ K → k2 ∈ K → ∃k3 : set, k3 ∈ K ∧ apply_fun phi k3 = apply_fun phi k1 ∧ apply_fun phi k3 = apply_fun phi k2) ∧ (∀k : set, k ∈ K → ∃j : set, j ∈ J ∧ apply_fun phi k = j ∧ apply_fun sub k = apply_fun net j) of type set → set → prop.
In Proofgold the corresponding term root is dc0586... and object id is 8b9626...
Definition. We define accumulation_point_of_net to be λX net x ⇒ ∃J X0 : set, directed_set J ∧ function_on net J X0 ∧ x ∈ X ∧ ∀U : set, x ∈ U → ∃i : set, i ∈ J ∧ apply_fun net i ∈ U ∧ apply_fun net i ≠ x of type set → set → set → prop.
In Proofgold the corresponding term root is 154733... and object id is 3485ef...
Definition. We define net_converges to be λX Tx net x ⇒ ∃J X0 : set, topology_on X Tx ∧ directed_set J ∧ function_on net J X0 ∧ x ∈ X ∧ ∀U : set, U ∈ Tx → x ∈ U → ∃i : set, i ∈ J ∧ apply_fun net i ∈ U of type set → set → set → set → prop.
In Proofgold the corresponding term root is 1cc009... and object id is 76d1de...
Theorem. (subnet_preserves_convergence)
∀X Tx net sub x : set, net_converges X Tx net x → subnet_of net sub → net_converges X Tx sub x
In Proofgold the corresponding term root is 3dcb65... and proposition id is afbbf1...
Proof:
Let X, Tx, net, sub and x be given.
Assume Hnet: net_converges X Tx net x.
Assume Hsub: subnet_of net sub.
We will prove net_converges X Tx sub x.
The rest of this subproof is missing.
∎
Theorem. (closure_via_nets)
∀X Tx A x : set, topology_on X Tx → (x ∈ closure_of X Tx A ↔ ∃net : set, net_on net ∧ net_converges X Tx net x)
In Proofgold the corresponding term root is 81fcea... and proposition id is 22c859...
Proof:
Let X, Tx, A and x be given.
Assume HTx: topology_on X Tx.
We will prove x ∈ closure_of X Tx A ↔ ∃net : set, net_on net ∧ net_converges X Tx net x.
The rest of this subproof is missing.
∎
Theorem. (continuity_via_nets)
∀X Tx Y Ty f : set, topology_on X Tx → topology_on Y Ty → (continuous_map X Tx Y Ty f ↔ ∀net : set, net_on net → ∀x : set, net_converges X Tx net x → net_converges Y Ty net (Empty))
In Proofgold the corresponding term root is 4b5777... and proposition id is c47315...
Proof:
Let X, Tx, Y, Ty and f be given.
Assume HTx: topology_on X Tx.
Assume HTy: topology_on Y Ty.
We will prove continuous_map X Tx Y Ty f ↔ ∀net : set, net_on net → ∀x : set, net_converges X Tx net x → net_converges Y Ty net (Empty).
The rest of this subproof is missing.
∎
Theorem. (subnet_converges_to_accumulation)
∀X Tx net x : set, accumulation_point_of_net X net x → ∃sub : set, subnet_of net sub ∧ net_converges X Tx sub x
In Proofgold the corresponding term root is 7ef1bc... and proposition id is 0f1a2e...
Proof:
Let X, Tx, net and x be given.
Assume Hacc: accumulation_point_of_net X net x.
We will prove ∃sub : set, subnet_of net sub ∧ net_converges X Tx sub x.
The rest of this subproof is missing.
∎
Theorem. (compact_iff_every_net_has_convergent_subnet)
∀X Tx : set, topology_on X Tx → (compact_space X Tx ↔ ∀net : set, net_on net → ∃sub x : set, subnet_of net sub ∧ net_converges X Tx sub x)
In Proofgold the corresponding term root is adf8cc... and proposition id is f7f4f5...
Proof:
Let X and Tx be given.
Assume HTx: topology_on X Tx.
We will prove compact_space X Tx ↔ ∀net : set, net_on net → ∃sub x : set, subnet_of net sub ∧ net_converges X Tx sub x.
The rest of this subproof is missing.
∎
Definition. We define countable_set to be λA ⇒ A ⊆ ω of type set → prop.
In Proofgold the corresponding term root is ee4423... and object id is 29fcb7...
Definition. We define countable_subcollection to be λV U ⇒ V ⊆ U ∧ countable_set V of type set → set → prop.
In Proofgold the corresponding term root is 49c45b... and object id is 615677...
Definition. We define countable_index_set to be λI ⇒ I ⊆ ω of type set → prop.
In Proofgold the corresponding term root is ee4423... and object id is 29fcb7...
Definition. We define countable_product_component_topology to be λXi i ⇒ apply_fun Xi i of type set → set → set.
In Proofgold the corresponding term root is f1f5ba... and object id is 0202ec...
Definition. We define real_sequences to be 𝒫 R of type set.
In Proofgold the corresponding term root is d9eef2... and object id is 572ecc...
Definition. We define uniform_metric_Romega to be Eps_i (λd ⇒ metric_on real_sequences d) of type set.
In Proofgold the corresponding term root is 25f623... and object id is 28267f...
Definition. We define uniform_topology to be metric_topology real_sequences uniform_metric_Romega of type set.
In Proofgold the corresponding term root is 5ef506... and object id is 12762a...
Definition. We define open_cover to be λX Tx U ⇒ (∀u : set, u ∈ U → u ∈ Tx) ∧ covers X U of type set → set → set → prop.
In Proofgold the corresponding term root is 2d2289... and object id is be938f...
Definition. We define Lindelof_space to be λX Tx ⇒ topology_on X Tx ∧ ∀U : set, open_cover X Tx U → ∃V : set, countable_subcollection V U ∧ covers X V of type set → set → prop.
In Proofgold the corresponding term root is 62e319... and object id is 260afc...
Definition. We define Sorgenfrey_line to be R of type set.
In Proofgold the corresponding term root is e26ffa... and object id is 5277ef...
Definition. We define Sorgenfrey_topology to be R_lower_limit_topology of type set.
In Proofgold the corresponding term root is d07fcd... and object id is 6262c6...
Definition. We define countable_basis_at to be λX Tx x ⇒ topology_on X Tx ∧ ∃B : set, basis_on X B ∧ countable_set B ∧ (∀U : set, U ∈ Tx → x ∈ U → ∃b : set, b ∈ B ∧ x ∈ b ∧ b ⊆ U) of type set → set → set → prop.
In Proofgold the corresponding term root is 346ee9... and object id is e2eade...
Definition. We define first_countable_space to be λX Tx ⇒ topology_on X Tx ∧ ∀x : set, x ∈ X → countable_basis_at X Tx x of type set → set → prop.
In Proofgold the corresponding term root is 13b6a7... and object id is 2aa2ad...
Theorem. (first_countable_sequences_detect_closure)
∀X Tx A x : set, topology_on X Tx → (∃seq : set, sequence_in seq A ∧ converges_to X Tx seq x) → x ∈ closure_of X Tx A
In Proofgold the corresponding term root is 93e7ae... and proposition id is 825eab...
Proof:
Let X, Tx, A and x be given.
Assume HTx: topology_on X Tx.
Assume Hseq: ∃seq : set, sequence_in seq A ∧ converges_to X Tx seq x.
We will prove x ∈ closure_of X Tx A.
The rest of this subproof is missing.
∎
Theorem. (first_countable_sequences_detect_continuity)
∀X Tx Y Ty f : set, topology_on X Tx → topology_on Y Ty → (continuous_map X Tx Y Ty f → ∀seq : set, sequence_in seq X → converges_to X Tx seq (Empty) → converges_to Y Ty (image_of f seq) f)
In Proofgold the corresponding term root is e50c5e... and proposition id is 3478f4...
Proof:
Let X, Tx, Y, Ty and f be given.
Assume HTx: topology_on X Tx.
Assume HTy: topology_on Y Ty.
We will prove continuous_map X Tx Y Ty f → ∀seq : set, sequence_in seq X → converges_to X Tx seq (Empty) → converges_to Y Ty (image_of f seq) f.
The rest of this subproof is missing.
∎
Definition. We define second_countable_space to be λX Tx ⇒ topology_on X Tx ∧ ∃B : set, basis_on X B ∧ countable_set B ∧ basis_generates X B Tx of type set → set → prop.
In Proofgold the corresponding term root is 46858b... and object id is 481362...
Theorem. (euclidean_spaces_second_countable)
∀n : set, second_countable_space (euclidean_space n) (euclidean_topology n)
In Proofgold the corresponding term root is b198d1... and proposition id is 1c7d66...
Proof:
Let n be given.
We will prove second_countable_space (euclidean_space n) (euclidean_topology n).
The rest of this subproof is missing.
∎
Theorem. (Romega_uniform_first_not_second_countable)
first_countable_space real_sequences uniform_topology ∧ ¬ second_countable_space real_sequences uniform_topology
In Proofgold the corresponding term root is d31ee6... and proposition id is 8f4d40...
Theorem. (countability_axioms_subspace_product)
∀X Tx : set, topology_on X Tx → (∀A : set, A ⊆ X → first_countable_space X Tx → first_countable_space A (subspace_topology X Tx A)) ∧ (∀A : set, A ⊆ X → second_countable_space X Tx → second_countable_space A (subspace_topology X Tx A)) ∧ (∀I Xi : set, countable_index_set I → (∀i : set, first_countable_space Xi (countable_product_component_topology Xi i)) → first_countable_space (countable_product_space I Xi) (countable_product_topology I Xi)) ∧ (∀I Xi : set, countable_index_set I → (∀i : set, second_countable_space Xi (countable_product_component_topology Xi i)) → second_countable_space (countable_product_space I Xi) (countable_product_topology I Xi))
In Proofgold the corresponding term root is 6cf0a8... and proposition id is 1b902f...
Proof:
Let X and Tx be given.
Assume HTx: topology_on X Tx.
We will prove (∀A : set, A ⊆ X → first_countable_space X Tx → first_countable_space A (subspace_topology X Tx A)) ∧ (∀A : set, A ⊆ X → second_countable_space X Tx → second_countable_space A (subspace_topology X Tx A)) ∧ (∀I Xi : set, countable_index_set I → (∀i : set, first_countable_space Xi (countable_product_component_topology Xi i)) → first_countable_space (countable_product_space I Xi) (countable_product_topology I Xi)) ∧ (∀I Xi : set, countable_index_set I → (∀i : set, second_countable_space Xi (countable_product_component_topology Xi i)) → second_countable_space (countable_product_space I Xi) (countable_product_topology I Xi)).
The rest of this subproof is missing.
∎
Definition. We define dense_in to be λA X Tx ⇒ closure_of X Tx A = X of type set → set → set → prop.
In Proofgold the corresponding term root is d0b42c... and object id is eb2621...
Theorem. (countable_basis_implies_Lindelof)
∀X Tx : set, topology_on X Tx → second_countable_space X Tx → ∀U : set, open_cover X Tx U → ∃V : set, countable_subcollection V U ∧ covers X V
In Proofgold the corresponding term root is bdcbba... and proposition id is 2d5d84...
Proof:
Let X and Tx be given.
Assume HTx: topology_on X Tx.
Assume Hscc: second_countable_space X Tx.
Let U be given.
Assume HU: open_cover X Tx U.
We will prove ∃V : set, countable_subcollection V U ∧ covers X V.
The rest of this subproof is missing.
∎
Theorem. (countable_basis_implies_separable)
∀X Tx : set, topology_on X Tx → second_countable_space X Tx → ∃D : set, countable_set D ∧ dense_in D X Tx
In Proofgold the corresponding term root is 7763dd... and proposition id is 641531...
Proof:
Let X and Tx be given.
Assume HTx: topology_on X Tx.
Assume Hscc: second_countable_space X Tx.
We will prove ∃D : set, countable_set D ∧ dense_in D X Tx.
The rest of this subproof is missing.
∎
Theorem. (Sorgenfrey_line_countability)
first_countable_space Sorgenfrey_line Sorgenfrey_topology ∧ dense_in rational_numbers Sorgenfrey_line Sorgenfrey_topology ∧ Lindelof_space Sorgenfrey_line Sorgenfrey_topology ∧ ¬ second_countable_space Sorgenfrey_line Sorgenfrey_topology
In Proofgold the corresponding term root is 2c8b79... and proposition id is 4c77e5...
Definition. We define Sorgenfrey_plane_topology to be product_topology Sorgenfrey_line Sorgenfrey_topology Sorgenfrey_line Sorgenfrey_topology of type set.
In Proofgold the corresponding term root is 96afb4... and object id is 5eec57...
Definition. We define one_point_sets_closed to be λX Tx ⇒ topology_on X Tx ∧ ∀x : set, x ∈ X → closed_in X Tx {x} of type set → set → prop.
In Proofgold the corresponding term root is 21cfa4... and object id is bd123f...
Definition. We define Hausdorff_spaces_family to be λI Xi ⇒ ∀i : set, i ∈ I → Hausdorff_space (product_component Xi i) (product_component_topology Xi i) of type set → set → prop.
In Proofgold the corresponding term root is 8ffab4... and object id is 8d711e...
Definition. We define regular_spaces_family to be λI Xi ⇒ ∀i : set, i ∈ I → topology_on (product_component Xi i) (product_component_topology Xi i) of type set → set → prop.
In Proofgold the corresponding term root is 7672d5... and object id is 1ea3ae...
Definition. We define uncountable_set to be λX ⇒ ¬ countable_set X of type set → prop.
In Proofgold the corresponding term root is a59567... and object id is 580e0b...
Definition. We define well_ordered_set to be λX ⇒ ∃alpha : set, ordinal alpha ∧ equip X alpha of type set → prop.
In Proofgold the corresponding term root is 6269e2... and object id is 3cdf9c...
Definition. We define completely_regular_spaces_family to be λI Xi ⇒ ∀i : set, i ∈ I → topology_on (product_component Xi i) (product_component_topology Xi i) of type set → set → prop.
In Proofgold the corresponding term root is 7672d5... and object id is 1ea3ae...
Definition. We define separating_family_of_functions to be λX Tx F J ⇒ topology_on X Tx ∧ F ⊆ function_space X J ∧ (∀x1 x2 : set, x1 ∈ X → x2 ∈ X → x1 ≠ x2 → ∃f : set, f ∈ F ∧ apply_fun f x1 ≠ apply_fun f x2) of type set → set → set → set → prop.
In Proofgold the corresponding term root is 9f5ca5... and object id is 1f16b6...
Definition. We define embedding_of to be λX Tx Y Ty f ⇒ function_on f X Y ∧ continuous_map X Tx Y Ty f ∧ (∀x1 x2 : set, x1 ∈ X → x2 ∈ X → apply_fun f x1 = apply_fun f x2 → x1 = x2) of type set → set → set → set → set → prop.
In Proofgold the corresponding term root is e5a1f7... and object id is 484014...
Definition. We define power_real to be λJ ⇒ function_space J R of type set → set.
In Proofgold the corresponding term root is 42af3b... and object id is bb8007...
Definition. We define unit_interval_power to be λJ ⇒ function_space J unit_interval of type set → set.
In Proofgold the corresponding term root is 42af3b... and object id is bb8007...
Definition. We define metrizable to be λX Tx ⇒ ∃d : set, metric_on X d ∧ metric_topology X d = Tx of type set → set → prop.
In Proofgold the corresponding term root is 69c21b... and object id is d040f9...
Theorem. (Sorgenfrey_plane_not_Lindelof)
¬ Lindelof_space (OrderedPair Sorgenfrey_line Sorgenfrey_line) Sorgenfrey_plane_topology
In Proofgold the corresponding term root is 195198... and proposition id is 8e9bc0...
Proof:
We will prove ¬ Lindelof_space (OrderedPair Sorgenfrey_line Sorgenfrey_line) Sorgenfrey_plane_topology.
The rest of this subproof is missing.
∎
Theorem. (ordered_square_subspace_not_Lindelof)
Lindelof_space ordered_square ordered_square_topology ∧ ¬ Lindelof_space ordered_square_open_strip ordered_square_subspace_topology
In Proofgold the corresponding term root is 123708... and proposition id is c39ac4...
Definition. We define regular_space to be λX Tx ⇒ topology_on X Tx ∧ ∀x : set, x ∈ X → ∀F : set, closed_in X Tx F → x ∉ F → ∃U V : set, U ∈ Tx ∧ V ∈ Tx ∧ x ∈ U ∧ F ⊆ V ∧ U ∩ V = Empty of type set → set → prop.
In Proofgold the corresponding term root is adfdb3... and object id is 3767ed...
Definition. We define normal_space to be λX Tx ⇒ topology_on X Tx ∧ ∀A B : set, closed_in X Tx A → closed_in X Tx B → A ∩ B = Empty → ∃U V : set, U ∈ Tx ∧ V ∈ Tx ∧ A ⊆ U ∧ B ⊆ V ∧ U ∩ V = Empty of type set → set → prop.
In Proofgold the corresponding term root is d78c03... and object id is c9d082...
Theorem. (regular_normal_via_closure)
∀X Tx : set, topology_on X Tx → (one_point_sets_closed X Tx → (regular_space X Tx ↔ ∀x U : set, x ∈ X → U ∈ Tx → x ∈ U → ∃V : set, V ∈ Tx ∧ x ∈ V ∧ closure_of X Tx V ⊆ U)) ∧ (one_point_sets_closed X Tx → (normal_space X Tx ↔ ∀A U : set, closed_in X Tx A → U ∈ Tx → A ⊆ U → ∃V : set, V ∈ Tx ∧ A ⊆ V ∧ closure_of X Tx V ⊆ U))
In Proofgold the corresponding term root is f9ac8f... and proposition id is 05bec1...
Proof:
Let X and Tx be given.
Assume HTx: topology_on X Tx.
We will prove (one_point_sets_closed X Tx → (regular_space X Tx ↔ ∀x U : set, x ∈ X → U ∈ Tx → x ∈ U → ∃V : set, V ∈ Tx ∧ x ∈ V ∧ closure_of X Tx V ⊆ U)) ∧ (one_point_sets_closed X Tx → (normal_space X Tx ↔ ∀A U : set, closed_in X Tx A → U ∈ Tx → A ⊆ U → ∃V : set, V ∈ Tx ∧ A ⊆ V ∧ closure_of X Tx V ⊆ U)).
The rest of this subproof is missing.
∎
Theorem. (separation_axioms_subspace_product)
∀X Tx : set, topology_on X Tx → (∀Y : set, Y ⊆ X → Hausdorff_space X Tx → Hausdorff_space Y (subspace_topology X Tx Y)) ∧ (∀I Xi : set, Hausdorff_spaces_family I Xi → Hausdorff_space (product_space I Xi) (product_topology_full I Xi)) ∧ (∀Y : set, Y ⊆ X → regular_space X Tx → regular_space Y (subspace_topology X Tx Y)) ∧ (∀I Xi : set, regular_spaces_family I Xi → regular_space (product_space I Xi) (product_topology_full I Xi))
In Proofgold the corresponding term root is fab340... and proposition id is ec6e3e...
Proof:
Let X and Tx be given.
Assume HTx: topology_on X Tx.
We will prove (∀Y : set, Y ⊆ X → Hausdorff_space X Tx → Hausdorff_space Y (subspace_topology X Tx Y)) ∧ (∀I Xi : set, Hausdorff_spaces_family I Xi → Hausdorff_space (product_space I Xi) (product_topology_full I Xi)) ∧ (∀Y : set, Y ⊆ X → regular_space X Tx → regular_space Y (subspace_topology X Tx Y)) ∧ (∀I Xi : set, regular_spaces_family I Xi → regular_space (product_space I Xi) (product_topology_full I Xi)).
The rest of this subproof is missing.
∎
Definition. We define R_K to be R of type set.
In Proofgold the corresponding term root is e26ffa... and object id is 5277ef...
Theorem. (RK_Hausdorff_not_regular)
Hausdorff_space R_K R_K_topology ∧ ¬ regular_space R_K R_K_topology
In Proofgold the corresponding term root is 4ec444... and proposition id is b6588e...
Proof:
We will prove Hausdorff_space R_K R_K_topology ∧ ¬ regular_space R_K R_K_topology.
The rest of this subproof is missing.
∎
Theorem. (Sorgenfrey_line_normal)
normal_space Sorgenfrey_line Sorgenfrey_topology
In Proofgold the corresponding term root is 333219... and proposition id is af1b4d...
Proof:
We will prove normal_space Sorgenfrey_line Sorgenfrey_topology.
The rest of this subproof is missing.
∎
Theorem. (Sorgenfrey_plane_not_normal)
regular_space (OrderedPair Sorgenfrey_line Sorgenfrey_line) Sorgenfrey_plane_topology ∧ ¬ normal_space (OrderedPair Sorgenfrey_line Sorgenfrey_line) Sorgenfrey_plane_topology
In Proofgold the corresponding term root is b93e1e... and proposition id is b6c1f0...
Theorem. (regular_countable_basis_normal)
∀X Tx : set, regular_space X Tx → second_countable_space X Tx → normal_space X Tx
In Proofgold the corresponding term root is 9d1ef4... and proposition id is 18bb49...
Proof:
Let X and Tx be given.
Assume Hreg: regular_space X Tx.
Assume Hscc: second_countable_space X Tx.
We will prove normal_space X Tx.
The rest of this subproof is missing.
∎
Theorem. (well_ordered_sets_normal)
∀X : set, well_ordered_set X → normal_space X (order_topology X)
In Proofgold the corresponding term root is bf0a78... and proposition id is 24ec63...
Proof:
Let X be given.
Assume Hwo: well_ordered_set X.
We will prove normal_space X (order_topology X).
The rest of this subproof is missing.
∎
Theorem. (metrizable_spaces_normal)
∀X d : set, metric_on X d → normal_space X (metric_topology X d)
In Proofgold the corresponding term root is 6a7d36... and proposition id is be6547...
Proof:
Let X and d be given.
Assume Hd: metric_on X d.
We will prove normal_space X (metric_topology X d).
The rest of this subproof is missing.
∎
Theorem. (compact_Hausdorff_normal)
∀X Tx : set, compact_space X Tx → Hausdorff_space X Tx → normal_space X Tx
In Proofgold the corresponding term root is 97cf3c... and proposition id is a4ff33...
Proof:
Let X and Tx be given.
Assume Hcomp: compact_space X Tx.
Assume HH: Hausdorff_space X Tx.
We will prove normal_space X Tx.
The rest of this subproof is missing.
∎
Theorem. (uncountable_product_R_not_normal)
∀J : set, uncountable_set J → ¬ normal_space (product_space J (const_family J R)) (product_topology_full J (const_family J R))
In Proofgold the corresponding term root is bc6dce... and proposition id is 440ce5...
Proof:
Let J be given.
Assume HJ: uncountable_set J.
We will prove ¬ normal_space (product_space J (const_family J R)) (product_topology_full J (const_family J R)).
The rest of this subproof is missing.
∎
Definition. We define S_Omega to be ω of type set.
In Proofgold the corresponding term root is 6fc30a... and object id is ad9ad8...
Definition. We define Sbar_Omega to be 𝒫 ω of type set.
In Proofgold the corresponding term root is a5ae09... and object id is f2e37c...
Definition. We define SOmega_topology to be discrete_topology S_Omega of type set.
In Proofgold the corresponding term root is a5ae09... and object id is f2e37c...
Definition. We define SbarOmega_topology to be discrete_topology Sbar_Omega of type set.
In Proofgold the corresponding term root is cc62c3... and object id is d51029...
Theorem. (SOmega_SbarOmega_not_normal)
normal_space S_Omega SOmega_topology ∧ normal_space Sbar_Omega SbarOmega_topology ∧ ¬ normal_space (product_space (OrderedPair S_Omega Sbar_Omega) (const_family (OrderedPair S_Omega Sbar_Omega) R)) (product_topology_full (OrderedPair S_Omega Sbar_Omega) (const_family (OrderedPair S_Omega Sbar_Omega) R))
In Proofgold the corresponding term root is e10781... and proposition id is 85c9dd...
Definition. We define closed_interval to be λa b ⇒ {x ∈ R|¬ (Rlt x a) ∧ ¬ (Rlt b x)} of type set → set → set.
In Proofgold the corresponding term root is 1149df... and object id is a7c019...
Theorem. (Urysohn_lemma)
∀X Tx A B a b : set, normal_space X Tx → closed_in X Tx A → closed_in X Tx B → A ∩ B = Empty → ∃f : set, continuous_map X Tx (closed_interval a b) (order_topology (closed_interval a b)) f
In Proofgold the corresponding term root is 83a6ac... and proposition id is 735eff...
Proof:
Let X, Tx, A, B, a and b be given.
Assume Hnorm: normal_space X Tx.
Assume HA: closed_in X Tx A.
Assume HB: closed_in X Tx B.
Assume Hdisj: A ∩ B = Empty.
We will prove ∃f : set, continuous_map X Tx (closed_interval a b) (order_topology (closed_interval a b)) f.
The rest of this subproof is missing.
∎
Definition. We define completely_regular_space to be λX Tx ⇒ topology_on X Tx ∧ ∀x : set, x ∈ X → ∀F : set, closed_in X Tx F → x ∉ F → ∃f : set, continuous_map X Tx R R_standard_topology f ∧ apply_fun f x = 0 ∧ ∀y : set, y ∈ F → apply_fun f y = 1 of type set → set → prop.
In Proofgold the corresponding term root is 51dbe4... and object id is 243d45...
Definition. We define Tychonoff_space to be λX Tx ⇒ completely_regular_space X Tx ∧ Hausdorff_space X Tx of type set → set → prop.
In Proofgold the corresponding term root is a39a77... and object id is 537884...
Theorem. (completely_regular_subspace_product)
∀X Tx : set, topology_on X Tx → (∀Y : set, Y ⊆ X → completely_regular_space X Tx → completely_regular_space Y (subspace_topology X Tx Y)) ∧ (∀I Xi : set, completely_regular_spaces_family I Xi → completely_regular_space (product_space I Xi) (product_topology_full I Xi))
In Proofgold the corresponding term root is 91ccec... and proposition id is f73ed6...
Proof:
Let X and Tx be given.
Assume HTx: topology_on X Tx.
We will prove (∀Y : set, Y ⊆ X → completely_regular_space X Tx → completely_regular_space Y (subspace_topology X Tx Y)) ∧ (∀I Xi : set, completely_regular_spaces_family I Xi → completely_regular_space (product_space I Xi) (product_topology_full I Xi)).
The rest of this subproof is missing.
∎
Theorem. (Sorgenfrey_plane_completely_regular_not_normal)
completely_regular_space (OrderedPair Sorgenfrey_line Sorgenfrey_line) Sorgenfrey_plane_topology ∧ ¬ normal_space (OrderedPair Sorgenfrey_line Sorgenfrey_line) Sorgenfrey_plane_topology
In Proofgold the corresponding term root is fdadd1... and proposition id is e12c36...
Theorem. (SOmega_SbarOmega_completely_regular_not_normal)
completely_regular_space (OrderedPair S_Omega Sbar_Omega) (product_topology_full (OrderedPair S_Omega Sbar_Omega) (const_family (OrderedPair S_Omega Sbar_Omega) R)) ∧ ¬ normal_space (OrderedPair S_Omega Sbar_Omega) (product_topology_full (OrderedPair S_Omega Sbar_Omega) (const_family (OrderedPair S_Omega Sbar_Omega) R))
In Proofgold the corresponding term root is e3798a... and proposition id is 499c26...
Theorem. (Urysohn_metrization_theorem)
∀X Tx : set, regular_space X Tx → second_countable_space X Tx → ∃d : set, metric_on X d ∧ metric_topology X d = Tx
In Proofgold the corresponding term root is 6fd825... and proposition id is 17a0ef...
Proof:
Let X and Tx be given.
Assume Hreg: regular_space X Tx.
Assume Hscc: second_countable_space X Tx.
We will prove ∃d : set, metric_on X d ∧ metric_topology X d = Tx.
The rest of this subproof is missing.
∎
Theorem. (embedding_via_functions)
∀X Tx : set, topology_on X Tx → one_point_sets_closed X Tx → ∀F J : set, separating_family_of_functions X Tx F J → ∃Fmap : set, embedding_of X Tx (power_real J) (product_topology_full J (const_family J R)) Fmap
In Proofgold the corresponding term root is 9c1a20... and proposition id is de6f34...
Proof:
Let X and Tx be given.
Assume HTx: topology_on X Tx.
Assume Hclosed: one_point_sets_closed X Tx.
Let F and J be given.
Assume Hsep: separating_family_of_functions X Tx F J.
We will prove ∃Fmap : set, embedding_of X Tx (power_real J) (product_topology_full J (const_family J R)) Fmap.
The rest of this subproof is missing.
∎
Theorem. (completely_regular_iff_embeds_in_cube)
∀X Tx : set, (completely_regular_space X Tx ↔ ∃J : set, ∃Fmap : set, embedding_of X Tx (unit_interval_power J) (product_topology_full J (const_family J unit_interval)) Fmap)
In Proofgold the corresponding term root is 7d376b... and proposition id is 503c92...
Proof:
Let X and Tx be given.
We will prove (completely_regular_space X Tx ↔ ∃J : set, ∃Fmap : set, embedding_of X Tx (unit_interval_power J) (product_topology_full J (const_family J unit_interval)) Fmap).
The rest of this subproof is missing.
∎
Theorem. (Tietze_extension_interval)
∀X Tx A a b f : set, normal_space X Tx → closed_in X Tx A → continuous_map A (subspace_topology X Tx A) (closed_interval a b) (order_topology (closed_interval a b)) f → ∃g : set, continuous_map X Tx (closed_interval a b) (order_topology (closed_interval a b)) g ∧ (∀x : set, x ∈ A → apply_fun g x = apply_fun f x)
In Proofgold the corresponding term root is f74b5e... and proposition id is df8666...
Proof:
Let X, Tx, A, a, b and f be given.
Assume Hnorm: normal_space X Tx.
Assume HA: closed_in X Tx A.
Assume Hf: continuous_map A (subspace_topology X Tx A) (closed_interval a b) (order_topology (closed_interval a b)) f.
We will prove ∃g : set, continuous_map X Tx (closed_interval a b) (order_topology (closed_interval a b)) g ∧ (∀x : set, x ∈ A → apply_fun g x = apply_fun f x).
The rest of this subproof is missing.
∎
Theorem. (Tietze_extension_real)
∀X Tx A f : set, normal_space X Tx → closed_in X Tx A → continuous_map A (subspace_topology X Tx A) R R_standard_topology f → ∃g : set, continuous_map X Tx R R_standard_topology g ∧ (∀x : set, x ∈ A → apply_fun g x = apply_fun f x)
In Proofgold the corresponding term root is 3d1406... and proposition id is b807c7...
Proof:
Let X, Tx, A and f be given.
Assume Hnorm: normal_space X Tx.
Assume HA: closed_in X Tx A.
Assume Hf: continuous_map A (subspace_topology X Tx A) R R_standard_topology f.
We will prove ∃g : set, continuous_map X Tx R R_standard_topology g ∧ (∀x : set, x ∈ A → apply_fun g x = apply_fun f x).
The rest of this subproof is missing.
∎
Definition. We define m_manifold to be λX Tx ⇒ Hausdorff_space X Tx ∧ second_countable_space X Tx of type set → set → prop.
In Proofgold the corresponding term root is c54efa... and object id is 0d00db...
Definition. We define partition_of_unity_dominated to be λX Tx U ⇒ topology_on X Tx ∧ open_cover X Tx U ∧ ∃P : set, P ⊆ function_space X R ∧ (∀f : set, f ∈ P → continuous_map X Tx R R_standard_topology f) ∧ (∀x : set, x ∈ X → ∃F : set, finite F ∧ F ⊆ P ∧ (∀f : set, f ∈ P → apply_fun f x ≠ 0 → f ∈ F) ∧ (∀f : set, f ∈ F → ∃u : set, u ∈ U ∧ {y ∈ X|apply_fun f y ≠ 0} ⊆ u)) of type set → set → set → prop.
In Proofgold the corresponding term root is dfe289... and object id is feb3dc...
Theorem. (finite_partition_of_unity_exists)
∀X Tx U : set, normal_space X Tx → finite U → open_cover X Tx U → ∃P : set, partition_of_unity_dominated X Tx U
In Proofgold the corresponding term root is da9f32... and proposition id is 77747c...
Proof:
Let X, Tx and U be given.
Assume Hnorm: normal_space X Tx.
Assume Hfin: finite U.
Assume Hcover: open_cover X Tx U.
We will prove ∃P : set, partition_of_unity_dominated X Tx U.
The rest of this subproof is missing.
∎
Theorem. (compact_manifold_embeds_in_Euclidean)
∀X Tx : set, m_manifold X Tx → compact_space X Tx → ∃N : set, ∃e : set, embedding_of X Tx (euclidean_space N) (euclidean_topology N) e
In Proofgold the corresponding term root is f5fa47... and proposition id is f4829c...
Proof:
Let X and Tx be given.
Assume Hman: m_manifold X Tx.
Assume Hcomp: compact_space X Tx.
We will prove ∃N : set, ∃e : set, embedding_of X Tx (euclidean_space N) (euclidean_topology N) e.
The rest of this subproof is missing.
∎
Theorem. (Tychonoff_theorem)
∀I Xi : set, (∀i : set, compact_space (product_component Xi i) (product_component_topology Xi i)) → compact_space (product_space I Xi) (product_topology_full I Xi)
In Proofgold the corresponding term root is 8f6e59... and proposition id is 8f729b...
Proof:
Let I and Xi be given.
Assume Hcomp: ∀i : set, compact_space (product_component Xi i) (product_component_topology Xi i).
We will prove compact_space (product_space I Xi) (product_topology_full I Xi).
The rest of this subproof is missing.
∎
Definition. We define Stone_Cech_compactification to be λX Tx ⇒ {p ∈ 𝒫 (𝒫 (𝒫 X))|∃Y Ty e : set, p = OrderedPair (OrderedPair Y Ty) e ∧ compact_space Y Ty ∧ Hausdorff_space Y Ty ∧ embedding_of X Tx Y Ty e} of type set → set → set.
In Proofgold the corresponding term root is e96a7d... and object id is 049d05...
Theorem. (Stone_Cech_universal_property)
∀X Tx : set, Tychonoff_space X Tx → compact_space (Stone_Cech_compactification X Tx) (Stone_Cech_compactification X Tx) ∧ Hausdorff_space (Stone_Cech_compactification X Tx) (Stone_Cech_compactification X Tx)
In Proofgold the corresponding term root is 76cf3a... and proposition id is b2e202...
Proof:
Let X and Tx be given.
Assume HT: Tychonoff_space X Tx.
We will prove compact_space (Stone_Cech_compactification X Tx) (Stone_Cech_compactification X Tx) ∧ Hausdorff_space (Stone_Cech_compactification X Tx) (Stone_Cech_compactification X Tx).
The rest of this subproof is missing.
∎
Definition. We define refine_of to be λV U ⇒ ∀v : set, v ∈ V → ∃u : set, u ∈ U ∧ v ⊆ u of type set → set → prop.
In Proofgold the corresponding term root is 784b8a... and object id is 2c5af6...
Definition. We define locally_finite_family to be λX Tx F ⇒ topology_on X Tx ∧ ∀x : set, x ∈ X → ∃N : set, N ∈ Tx ∧ x ∈ N ∧ ∃S : set, finite S ∧ S ⊆ F ∧ ∀A : set, A ∈ F → A ∩ N ≠ Empty → A ∈ S of type set → set → set → prop.
In Proofgold the corresponding term root is 39b973... and object id is 0eacf7...
Definition. We define locally_finite_basis to be λX Tx ⇒ topology_on X Tx ∧ ∃B : set, basis_on X B ∧ locally_finite_family X Tx B of type set → set → prop.
In Proofgold the corresponding term root is 0c9cfd... and object id is 4100f2...
Definition. We define sigma_locally_finite_basis to be λX Tx ⇒ topology_on X Tx ∧ ∃Fams : set, countable_set Fams ∧ Fams ⊆ 𝒫 (𝒫 X) ∧ (∀F : set, F ∈ Fams → locally_finite_family X Tx F) ∧ basis_on X (⋃ Fams) ∧ ∀b : set, b ∈ ⋃ Fams → b ∈ Tx of type set → set → prop.
In Proofgold the corresponding term root is 9c0f09... and object id is 13bcec...
Theorem. (Nagata_Smirnov_metrization)
∀X Tx : set, regular_space X Tx → sigma_locally_finite_basis X Tx → metrizable X Tx
In Proofgold the corresponding term root is 2883d2... and proposition id is 78a6e9...
Proof:
Let X and Tx be given.
Assume Hreg: regular_space X Tx.
Assume Hbasis: sigma_locally_finite_basis X Tx.
We will prove metrizable X Tx.
The rest of this subproof is missing.
∎
Definition. We define paracompact_space to be λX Tx ⇒ topology_on X Tx ∧ ∀U : set, open_cover X Tx U → ∃V : set, open_cover X Tx V ∧ locally_finite_family X Tx V ∧ refine_of V U of type set → set → prop.
In Proofgold the corresponding term root is 54507e... and object id is 3c02cb...
Theorem. (locally_finite_refinement)
∀X Tx U : set, paracompact_space X Tx → open_cover X Tx U → ∃V : set, open_cover X Tx V ∧ locally_finite_family X Tx V
In Proofgold the corresponding term root is a526c8... and proposition id is de5dbd...
Proof:
Let X, Tx and U be given.
Assume Hpara: paracompact_space X Tx.
Assume Hcover: open_cover X Tx U.
We will prove ∃V : set, open_cover X Tx V ∧ locally_finite_family X Tx V.
The rest of this subproof is missing.
∎
Theorem. (paracompact_Hausdorff_normal)
∀X Tx : set, paracompact_space X Tx → Hausdorff_space X Tx → normal_space X Tx
In Proofgold the corresponding term root is b8c5c3... and proposition id is 0cee32...
Proof:
Let X and Tx be given.
Assume Hpara: paracompact_space X Tx.
Assume HH: Hausdorff_space X Tx.
We will prove normal_space X Tx.
The rest of this subproof is missing.
∎
Theorem. (Smirnov_metrization)
∀X Tx : set, regular_space X Tx → locally_finite_basis X Tx → metrizable X Tx
In Proofgold the corresponding term root is 727a74... and proposition id is 46be5b...
Proof:
Let X and Tx be given.
Assume Hreg: regular_space X Tx.
Assume Hbasis: locally_finite_basis X Tx.
We will prove metrizable X Tx.
The rest of this subproof is missing.
∎
Definition. We define cauchy_sequence to be λX d seq ⇒ metric_on X d ∧ seq ⊆ X ∧ ∀eps : set, eps ∈ R → ∃N : set, N ∈ ω ∧ ∀m n : set, m ∈ ω → n ∈ ω → N ⊆ ω → Rlt (d (apply_fun seq m) (apply_fun seq n)) eps of type set → set → set → prop.
In Proofgold the corresponding term root is 5c95dc... and object id is 86faf2...
Definition. We define complete_metric_space to be λX d ⇒ metric_on X d ∧ ∀seq : set, seq ⊆ X → cauchy_sequence X d seq → ∃x : set, converges_to X (metric_topology X d) seq x of type set → set → prop.
In Proofgold the corresponding term root is e7ff6a... and object id is 063d1b...
Definition. We define discrete_metric to be λX ⇒ {p ∈ OrderedPair X X|∃x : set, ∃y : set, x ∈ X ∧ y ∈ X ∧ ((x = y ∧ p = OrderedPair (OrderedPair x y) 0) ∨ (x ≠ y ∧ p = OrderedPair (OrderedPair x y) 1))} of type set → set.
In Proofgold the corresponding term root is f259c5... and object id is 0b8bc3...
Definition. We define euclidean_metric to be λn ⇒ discrete_metric (euclidean_space n) of type set → set.
In Proofgold the corresponding term root is 4b052b... and object id is a49d44...
Definition. We define bounded_product_metric to be λJ ⇒ discrete_metric (power_real J) of type set → set.
In Proofgold the corresponding term root is 0cd401... and object id is 2feacd...
Theorem. (Cauchy_with_convergent_subsequence_converges)
∀X d seq x : set, metric_on X d → cauchy_sequence X d seq → (∃subseq : set, subseq ⊆ seq ∧ converges_to X (metric_topology X d) subseq x) → converges_to X (metric_topology X d) seq x
In Proofgold the corresponding term root is 317abb... and proposition id is 2b4257...
Proof:
Let X, d, seq and x be given.
Assume Hd: metric_on X d.
Assume Hcauchy: cauchy_sequence X d seq.
Assume Hsub: ∃subseq : set, subseq ⊆ seq ∧ converges_to X (metric_topology X d) subseq x.
We will prove converges_to X (metric_topology X d) seq x.
The rest of this subproof is missing.
∎
Theorem. (Euclidean_space_complete)
∀k : set, complete_metric_space (euclidean_space k) (euclidean_metric k)
In Proofgold the corresponding term root is 97858d... and proposition id is 9e3da6...
Proof:
Let k be given.
We will prove complete_metric_space (euclidean_space k) (euclidean_metric k).
The rest of this subproof is missing.
∎
Theorem. (product_sequence_convergence_iff_coordinates)
∀X J : set, X = product_space J (const_family J R) → ∀seq x : set, converges_to X (product_topology_full J (const_family J R)) seq x ↔ (∀j : set, j ∈ J → converges_to (product_component (const_family J R) j) (product_component_topology (const_family J R) j) (Repl seq (λs ⇒ apply_fun s j)) (apply_fun x j))
In Proofgold the corresponding term root is 86fa24... and proposition id is 801ec0...
Proof:
Let X and J be given.
Assume HX: X = product_space J (const_family J R).
Let seq and x be given.
We will prove converges_to X (product_topology_full J (const_family J R)) seq x ↔ (∀j : set, j ∈ J → converges_to (product_component (const_family J R) j) (product_component_topology (const_family J R) j) (Repl seq (λs ⇒ apply_fun s j)) (apply_fun x j)).
The rest of this subproof is missing.
∎
Theorem. (product_Romega_complete)
complete_metric_space (power_real ω) (bounded_product_metric ω)
In Proofgold the corresponding term root is 39c3f2... and proposition id is d4fab4...
Proof:
We will prove complete_metric_space (power_real ω) (bounded_product_metric ω).
The rest of this subproof is missing.
∎
Definition. We define unit_square to be OrderedPair unit_interval unit_interval of type set.
In Proofgold the corresponding term root is 60f7e1... and object id is d6fb09...
Definition. We define unit_square_topology to be product_topology unit_interval R_standard_topology unit_interval R_standard_topology of type set.
In Proofgold the corresponding term root is e34d9f... and object id is 2642d2...
Theorem. (space_filling_curve)
∃f : set, continuous_map unit_interval R2_standard_topology unit_square unit_square_topology f
In Proofgold the corresponding term root is 34ed6e... and proposition id is 17bc5a...
Proof:
We will prove ∃f : set, continuous_map unit_interval R2_standard_topology unit_square unit_square_topology f.
The rest of this subproof is missing.
∎
Definition. We define sequentially_compact to be λX Tx ⇒ topology_on X Tx ∧ ∀seq : set, seq ⊆ X → ∃x : set, converges_to X Tx seq x of type set → set → prop.
In Proofgold the corresponding term root is cc88d9... and object id is 4eb7fe...
Theorem. (compact_metric_equivalences)
∀X d : set, metric_on X d → (compact_space X (metric_topology X d) ↔ sequentially_compact X (metric_topology X d))
In Proofgold the corresponding term root is 744f62... and proposition id is cecd9e...
Proof:
Let X and d be given.
Assume Hd: metric_on X d.
We will prove compact_space X (metric_topology X d) ↔ sequentially_compact X (metric_topology X d).
The rest of this subproof is missing.
∎
Definition. We define pointwise_convergence_topology to be λX Tx Y Ty ⇒ generated_topology (function_space X Y) Empty of type set → set → set → set → set.
In Proofgold the corresponding term root is 2565fa... and object id is 948af7...
Definition. We define compact_convergence_topology to be λX Tx Y Ty ⇒ generated_topology (function_space X Y) Empty of type set → set → set → set → set.
In Proofgold the corresponding term root is 2565fa... and object id is 948af7...
Definition. We define equicontinuous_family to be λX Tx Y Ty F ⇒ topology_on X Tx ∧ topology_on Y Ty ∧ F ⊆ function_space X Y ∧ ∀x : set, x ∈ X → ∀V : set, V ∈ Ty → (∃f0 : set, f0 ∈ F ∧ apply_fun f0 x ∈ V) → ∃U : set, U ∈ Tx ∧ x ∈ U ∧ ∀f : set, f ∈ F → ∀y : set, y ∈ U → apply_fun f y ∈ V of type set → set → set → set → set → prop.
In Proofgold the corresponding term root is 32f23f... and object id is 8b040e...
Definition. We define relatively_compact_in_compact_convergence to be λX Tx Y Ty F ⇒ topology_on X Tx ∧ topology_on Y Ty ∧ F ⊆ function_space X Y ∧ compact_space F (compact_convergence_topology X Tx Y Ty) of type set → set → set → set → set → prop.
In Proofgold the corresponding term root is 05f692... and object id is a54e1b...
Theorem. (Ascoli_theorem)
∀X Tx Y Ty F : set, compact_space X Tx → Hausdorff_space Y Ty → equicontinuous_family X Tx Y Ty F → relatively_compact_in_compact_convergence X Tx Y Ty F
In Proofgold the corresponding term root is ba48aa... and proposition id is 976730...
Proof:
Let X, Tx, Y, Ty and F be given.
Assume Hcomp: compact_space X Tx.
Assume HH: Hausdorff_space Y Ty.
Assume Heq: equicontinuous_family X Tx Y Ty F.
We will prove relatively_compact_in_compact_convergence X Tx Y Ty F.
The rest of this subproof is missing.
∎
Definition. We define intersection_over_family to be λX Fam ⇒ {x ∈ X|∀U : set, U ∈ Fam → x ∈ U} of type set → set → set.
In Proofgold the corresponding term root is 6d7088... and object id is 443aac...
Definition. We define Baire_space to be λTx ⇒ ∃X : set, topology_on X Tx ∧ ∀U : set, U ⊆ Tx → countable_set U → (∀u : set, u ∈ U → u ∈ Tx ∧ dense_in u X Tx) → dense_in (intersection_over_family X U) X Tx of type set → prop.
In Proofgold the corresponding term root is db69fc... and object id is 5027f2...
Theorem. (Baire_space_dense_Gdelta)
∀Tx : set, (Baire_space Tx ↔ ∃X : set, topology_on X Tx ∧ ∀U : set, U ⊆ Tx → countable_set U → (∀u : set, u ∈ U → u ∈ Tx ∧ dense_in u X Tx) → dense_in (intersection_over_family X U) X Tx)
In Proofgold the corresponding term root is 4930ef... and proposition id is 4f9051...
Proof:
Let Tx be given.
We will prove Baire_space Tx ↔ (∃X : set, topology_on X Tx ∧ ∀U : set, U ⊆ Tx → countable_set U → (∀u : set, u ∈ U → u ∈ Tx ∧ dense_in u X Tx) → dense_in (intersection_over_family X U) X Tx).
The rest of this subproof is missing.
∎
Theorem. (Baire_category_complete_metric)
∀X d : set, complete_metric_space X d → Baire_space (metric_topology X d)
In Proofgold the corresponding term root is 900e83... and proposition id is 68e900...
Proof:
Let X and d be given.
Assume Hcomp.
We will prove Baire_space (metric_topology X d).
The rest of this subproof is missing.
∎
Theorem. (Baire_category_compact_Hausdorff)
∀X Tx : set, compact_space X Tx → Hausdorff_space X Tx → Baire_space Tx
In Proofgold the corresponding term root is c0a1d8... and proposition id is 61d2c1...
Proof:
Let X and Tx be given.
Assume Hcomp HHaus.
We will prove Baire_space Tx.
The rest of this subproof is missing.
∎
Theorem. (Baire_category_theorem)
∀X : set, Baire_space X → ∀U : set, open_in X X U → U ≠ Empty
In Proofgold the corresponding term root is a087c7... and proposition id is f634cd...
Proof:
Let X be given.
Assume HBaire.
Let U be given.
Assume Hopen.
We will prove U ≠ Empty.
The rest of this subproof is missing.
∎
Definition. We define differentiable_at to be λf x ⇒ False of type set → set → prop.
In Proofgold the corresponding term root is d1fb07... and object id is ecc99e...
Definition. We define nowhere_differentiable to be λf ⇒ function_on f R R ∧ ∀x : set, x ∈ R → ¬ differentiable_at f x of type set → prop.
In Proofgold the corresponding term root is b4c6eb... and object id is 9af630...
Theorem. (nowhere_differentiable_function_exists)
∃f : set, continuous_map R R_standard_topology R R_standard_topology f ∧ nowhere_differentiable f
In Proofgold the corresponding term root is b1bed8... and proposition id is 5771b6...
Proof:
We will prove ∃f : set, continuous_map R R_standard_topology R R_standard_topology f ∧ nowhere_differentiable f.
The rest of this subproof is missing.
∎
Definition. We define cardinality_exact to be λS n ⇒ ordinal n ∧ equip S n of type set → set → prop.
In Proofgold the corresponding term root is b2404a... and object id is 71475a...
Definition. We define cardinality_at_most to be λS n ⇒ ordinal n ∧ ∃k : set, ordinal k ∧ k ⊆ n ∧ equip S k of type set → set → prop.
In Proofgold the corresponding term root is 1a6739... and object id is a55e8a...
Definition. We define collection_has_order_at_m_plus_one to be λX A m ⇒ ordinal m ∧ (∃x : set, x ∈ X ∧ ∃Fam : set, Fam ⊆ A ∧ finite Fam ∧ cardinality_exact Fam m ∧ ∀U : set, U ∈ Fam → x ∈ U) ∧ ∀x : set, x ∈ X → cardinality_at_most {U ∈ A|x ∈ U} m of type set → set → set → prop.
In Proofgold the corresponding term root is 40f43c... and object id is 085d57...
Definition. We define covering_dimension to be λX n ⇒ n ∈ ω ∧ ∃Tx : set, topology_on X Tx of type set → set → prop.
In Proofgold the corresponding term root is a18356... and object id is c001c9...
Definition. We define finite_dimensional_space to be λX Tx ⇒ topology_on X Tx ∧ ∃m : set, covering_dimension X m of type set → set → prop.
In Proofgold the corresponding term root is 9763f5... and object id is 60c5ff...
Theorem. (covering_dimension_properties)
∀X : set, ∃n : set, covering_dimension X n
In Proofgold the corresponding term root is 4add03... and proposition id is 6580c1...
Proof:
Let X be given.
We use Empty to witness the existential quantifier.
The rest of this subproof is missing.
∎
Theorem. (compact_subspace_Rn_dimension_le)
∀X n : set, compact_space X (euclidean_topology n) → covering_dimension X n
In Proofgold the corresponding term root is 9b7617... and proposition id is 616904...
Proof:
Let X and n be given.
Assume Hcomp.
We will prove covering_dimension X n.
The rest of this subproof is missing.
∎
Theorem. (compact_manifold_dimension_le)
∀X Tx m : set, m_manifold X Tx → compact_space X Tx → covering_dimension X m
In Proofgold the corresponding term root is e1f818... and proposition id is 13ad18...
Proof:
Let X, Tx and m be given.
Assume Hman Hcomp.
We will prove covering_dimension X m.
The rest of this subproof is missing.
∎
Theorem. (Menger_Nobeling_embedding)
∀X Tx m : set, compact_space X Tx → metrizable X Tx → covering_dimension X m → ∃N : set, ∃e : set, embedding_of X Tx (euclidean_space N) (euclidean_topology N) e
In Proofgold the corresponding term root is 2ece4b... and proposition id is 503383...
Proof:
Let X, Tx and m be given.
Assume Hcomp Hmet Hdim.
We will prove ∃N : set, ∃e : set, embedding_of X Tx (euclidean_space N) (euclidean_topology N) e.
The rest of this subproof is missing.
∎
Theorem. (dimension_closed_subspace_le)
∀X Tx Y n : set, covering_dimension X n → closed_in X Tx Y → covering_dimension Y n
In Proofgold the corresponding term root is b40f35... and proposition id is 2888f5...
Proof:
Let X, Tx, Y and n be given.
Assume HX HY.
We will prove covering_dimension Y n.
The rest of this subproof is missing.
∎
Theorem. (dimension_union_closed_max)
∀X Y Z n : set, covering_dimension Y n → covering_dimension Z n → covering_dimension (Y ∪ Z) n
In Proofgold the corresponding term root is e0e360... and proposition id is 8981ea...
Proof:
Let X, Y, Z and n be given.
Assume HY HZ.
We will prove covering_dimension (Y ∪ Z) n.
We will prove n ∈ ω ∧ ∃Tx : set, topology_on (Y ∪ Z) Tx.
Apply HY to the current goal.
Assume Hn HTy.
Apply andI to the current goal.
An exact proof term for the current goal is Hn.
The rest of this subproof is missing.
∎
Theorem. (dimension_finite_union_closed_max)
∀X Fam n : set, finite Fam → (∀Y : set, Y ∈ Fam → covering_dimension Y n) → covering_dimension (⋃ Fam) n
In Proofgold the corresponding term root is 65a8d3... and proposition id is 1eeabb...
Proof:
Let X, Fam and n be given.
Assume Hfin Hall.
We will prove covering_dimension (⋃ Fam) n.
The rest of this subproof is missing.
∎
Theorem. (compact_1_manifold_dimension_1)
∀X Tx : set, compact_space X Tx → m_manifold X Tx → covering_dimension X (Sing Empty)
In Proofgold the corresponding term root is 78178d... and proposition id is 506b8f...
Proof:
Let X and Tx be given.
Assume Hcomp Hman.
The rest of this subproof is missing.
∎
Definition. We define two to be Sing (Sing Empty) of type set.
In Proofgold the corresponding term root is 4145a1... and object id is 07be9b...
Theorem. (compact_2_manifold_dimension_le_2)
∀X Tx : set, compact_space X Tx → m_manifold X Tx → covering_dimension X two
In Proofgold the corresponding term root is 224b4c... and proposition id is 614d2c...
Proof:
Let X and Tx be given.
Assume Hcomp Hman.
The rest of this subproof is missing.
∎
Definition. We define arc to be λX Tx ⇒ ∃f : set, homeomorphism unit_interval R_standard_topology X Tx f of type set → set → prop.
In Proofgold the corresponding term root is 49191d... and object id is 1f3d93...
Theorem. (arc_is_topological_space)
∀X Tx : set, arc X Tx → topology_on X Tx
In Proofgold the corresponding term root is d1c7a5... and proposition id is 8e3662...
Proof:
Let X and Tx be given.
Assume Harc.
Apply Harc to the current goal.
Let f be given.
Assume Hhom.
The rest of this subproof is missing.
∎
Definition. We define end_points_of_arc to be λX Tx p q ⇒ arc X Tx ∧ p ∈ X ∧ q ∈ X ∧ p ≠ q ∧ connected_space (X ∖ (Sing p)) Tx ∧ connected_space (X ∖ (Sing q)) Tx of type set → set → set → set → prop.
In Proofgold the corresponding term root is 401aa6... and object id is 48298c...
Definition. We define linear_graph to be λG Tg ⇒ Hausdorff_space G Tg ∧ ∃Arcs : set, finite Arcs ∧ (∀A : set, A ∈ Arcs → ∃Ta : set, arc A Ta ∧ A ⊆ G) ∧ G = ⋃ Arcs ∧ (∀A B : set, A ∈ Arcs → B ∈ Arcs → A ≠ B → ∃p : set, (A ∩ B = Empty ∨ A ∩ B = Sing p)) of type set → set → prop.
In Proofgold the corresponding term root is ee06cb... and object id is cc6a01...
Theorem. (linear_graph_dimension_1)
∀G Tg : set, linear_graph G Tg → covering_dimension G (Sing Empty)
In Proofgold the corresponding term root is 10fa97... and proposition id is 1db20e...
Proof:
Let G and Tg be given.
Assume Hlin.
The rest of this subproof is missing.
∎
Definition. We define collinear_in_R3 to be λp q r ⇒ p ∈ R ∧ q ∈ R ∧ r ∈ R ∧ ∃t1 t2 : set, t1 ∈ R ∧ t2 ∈ R ∧ (∃a b : set, a ∈ R ∧ b ∈ R ∧ r = a ∖ (b ∖ (p ∖ (q ∖ p)))) of type set → set → set → prop.
In Proofgold the corresponding term root is de0517... and object id is ca70f2...
Definition. We define coplanar_in_R3 to be λp q r s ⇒ p ∈ R ∧ q ∈ R ∧ r ∈ R ∧ s ∈ R ∧ ∃A B C D : set, A ∈ R ∧ B ∈ R ∧ C ∈ R ∧ D ∈ R ∧ True of type set → set → set → set → prop.
In Proofgold the corresponding term root is 7d6f64... and object id is 99bff2...
Definition. We define geometrically_independent to be λS ⇒ S ⊆ R of type set → prop.
In Proofgold the corresponding term root is b26a40... and object id is 2703f1...
Definition. We define affine_plane to be λS ⇒ {x ∈ R|∃tcoeffs : set, (∀s : set, s ∈ S → ∃t : set, t ∈ R ∧ (s,t) ∈ tcoeffs) ∧ True} of type set → set.
In Proofgold the corresponding term root is c2a74d... and object id is 9b9d95...
Definition. We define k_plane to be λk P ⇒ k ∈ ω ∧ ∃S : set, geometrically_independent S ∧ finite S ∧ (∃kp1 : set, kp1 = k ∪ (Sing k) ∧ equip S kp1) ∧ P = affine_plane S of type set → set → prop.
In Proofgold the corresponding term root is 8e607d... and object id is eace96...
Definition. We define general_position_RN to be λN A ⇒ N ∈ ω ∧ A ⊆ R ∧ ∀S : set, S ⊆ A → (∀Np1 : set, Np1 = N ∪ (Sing N) → (∃f : set → set, inj S Np1 f) → geometrically_independent S) of type set → set → prop.
In Proofgold the corresponding term root is cc1422... and object id is 3e546f...
Theorem. (finite_set_approximation_general_position)
∀N : set, ∀pts : set, ∀delta : set, N ∈ ω → finite pts → pts ⊆ R → delta ∈ R → ∃pts' : set, general_position_RN N pts' ∧ finite pts' ∧ equip pts pts'
In Proofgold the corresponding term root is 059f4d... and proposition id is c73f52...
Proof:
Let N, pts and delta be given.
Assume HN Hfin Hpts Hdelta.
We will prove ∃pts' : set, general_position_RN N pts' ∧ finite pts' ∧ equip pts pts'.
The rest of this subproof is missing.
∎
Theorem. (Menger_Nobeling_embedding_full)
∀X Tx m : set, compact_space X Tx → metrizable X Tx → covering_dimension X m → m ∈ ω → ∃N : set, ∃e : set, N = m ∪ m ∪ (Sing Empty) ∧ embedding_of X Tx (euclidean_space N) (euclidean_topology N) e
In Proofgold the corresponding term root is 07983b... and proposition id is b123c2...
Proof:
Let X, Tx and m be given.
Assume Hcomp Hmet Hdim Hm.
We will prove ∃N : set, ∃e : set, N = m ∪ m ∪ (Sing Empty) ∧ embedding_of X Tx (euclidean_space N) (euclidean_topology N) e.
The rest of this subproof is missing.
∎
Theorem. (compact_subspace_RN_dimension_le_N)
∀X N : set, N ∈ ω → compact_space X (euclidean_topology N) → X ⊆ (euclidean_space N) → covering_dimension X N
In Proofgold the corresponding term root is 7f1848... and proposition id is bf402a...
Proof:
Let X and N be given.
Assume HN Hcomp Hsub.
We will prove covering_dimension X N.
The rest of this subproof is missing.
∎
Theorem. (compact_m_manifold_dimension_le_m)
∀X Tx m : set, m ∈ ω → compact_space X Tx → m_manifold X Tx → covering_dimension X m
In Proofgold the corresponding term root is abd24b... and proposition id is f74d24...
Proof:
Let X, Tx and m be given.
Assume Hm Hcomp Hman.
We will prove covering_dimension X m.
The rest of this subproof is missing.
∎
Theorem. (compact_m_manifold_embeds_R2mp1)
∀X Tx m : set, m ∈ ω → compact_space X Tx → m_manifold X Tx → ∃N : set, ∃e : set, N = m ∪ m ∪ (Sing Empty) ∧ embedding_of X Tx (euclidean_space N) (euclidean_topology N) e
In Proofgold the corresponding term root is 4d60ec... and proposition id is 0c7197...
Proof:
Let X, Tx and m be given.
Assume Hm Hcomp Hman.
We will prove ∃N : set, ∃e : set, N = m ∪ m ∪ (Sing Empty) ∧ embedding_of X Tx (euclidean_space N) (euclidean_topology N) e.
The rest of this subproof is missing.
∎
Theorem. (compact_metrizable_embeds_iff_finite_dim)
∀X Tx : set, compact_space X Tx → metrizable X Tx → ((∃N : set, ∃e : set, N ∈ ω ∧ embedding_of X Tx (euclidean_space N) (euclidean_topology N) e) ↔ finite_dimensional_space X Tx)
In Proofgold the corresponding term root is 36f661... and proposition id is 573b25...
Proof:
Let X and Tx be given.
Assume HC: compact_space X Tx.
Assume HM: metrizable X Tx.
We will prove (∃N : set, ∃e : set, N ∈ ω ∧ embedding_of X Tx (euclidean_space N) (euclidean_topology N) e) ↔ finite_dimensional_space X Tx.
The rest of this subproof is missing.
∎
Definition. We define locally_m_euclidean to be λX Tx m ⇒ m ∈ ω ∧ topology_on X Tx ∧ ∀x : set, x ∈ X → ∃U : set, ∃V : set, ∃f : set, open_in X Tx U ∧ x ∈ U ∧ V ⊆ (euclidean_space m) ∧ open_in (euclidean_space m) (euclidean_topology m) V ∧ homeomorphism U (subspace_topology X Tx U) V (subspace_topology (euclidean_space m) (euclidean_topology m) V) f of type set → set → set → prop.
In Proofgold the corresponding term root is 50e288... and object id is 60135c...
Theorem. (ex50_1_discrete_dimension_0)
∀X Tx : set, Tx = discrete_topology X → topology_on X Tx → covering_dimension X Empty
In Proofgold the corresponding term root is 2058e3... and proposition id is 727ad3...
Proof:
Let X and Tx be given.
Assume HTxdisc HTxtop.
The rest of this subproof is missing.
∎
Theorem. (ex50_2_connected_T1_dimension_ge_1)
∀X Tx : set, connected_space X Tx → T1_space X Tx → (∃x y : set, x ∈ X ∧ y ∈ X ∧ x ≠ y) → covering_dimension X Empty → False
In Proofgold the corresponding term root is 7aaacc... and proposition id is 24c9aa...
Proof:
Let X and Tx be given.
Assume Hconn HT1 Hdist Hdim0.
We will prove False.
The rest of this subproof is missing.
∎
Theorem. (ex50_3_sine_curve_dimension_1)
∀X Tx : set, X = R → covering_dimension X (Sing Empty)
In Proofgold the corresponding term root is eb7cab... and proposition id is f90389...
Proof:
Let X and Tx be given.
Assume HX.
We will prove covering_dimension X (Sing Empty).
The rest of this subproof is missing.
∎
Theorem. (ex50_4_points_general_position_R3)
∀zero e1 e2 e3 ones : set, zero = R → e1 = R → e2 = R → e3 = R → ones = R → general_position_RN (Sing (Sing (Sing Empty))) {zero,e1,e2,e3,ones}
In Proofgold the corresponding term root is 544954... and proposition id is e4ee2b...
Proof:
Let zero, e1, e2, e3 and ones be given.
Assume Hz He1 He2 He3 Hones.
We will prove general_position_RN (Sing (Sing (Sing Empty))) {zero,e1,e2,e3,ones}.
The rest of this subproof is missing.
∎
Theorem. (ex50_5_embedding_m1_linear_graph)
∀X Tx : set, covering_dimension X (Sing Empty) → compact_space X Tx → metrizable X Tx → ∃g : set, (∀x : set, x ∈ X → apply_fun g x ∈ (euclidean_space (Sing (Sing (Sing Empty))))) ∧ linear_graph (apply_fun g X) R_standard_topology
In Proofgold the corresponding term root is 819a52... and proposition id is 823582...
Proof:
Let X and Tx be given.
Assume Hdim Hcomp Hmet.
We will prove ∃g : set, (∀x : set, x ∈ X → apply_fun g x ∈ (euclidean_space (Sing (Sing (Sing Empty))))) ∧ linear_graph (apply_fun g X) R_standard_topology.
The rest of this subproof is missing.
∎
Theorem. (ex50_6_locally_compact_embeds)
∀X Tx m : set, m ∈ ω → locally_compact X Tx → Hausdorff_space X Tx → second_countable_space X Tx → (∀C : set, C ⊆ X → compact_space C (subspace_topology X Tx C) → covering_dimension C m) → ∃N : set, ∃e : set, N = m ∪ m ∪ (Sing Empty) ∧ embedding_of X Tx (euclidean_space N) (euclidean_topology N) e ∧ closed_in (euclidean_space N) (euclidean_topology N) (apply_fun e X)
In Proofgold the corresponding term root is 63fab7... and proposition id is ffdc55...
Proof:
Let X, Tx and m be given.
Assume Hm Hlc HHaus Hsec Hdim.
We will prove ∃N : set, ∃e : set, N = m ∪ m ∪ (Sing Empty) ∧ embedding_of X Tx (euclidean_space N) (euclidean_topology N) e ∧ closed_in (euclidean_space N) (euclidean_topology N) (apply_fun e X).
The rest of this subproof is missing.
∎
Theorem. (ex50_7_manifold_closed_embedding)
∀X Tx m : set, m ∈ ω → m_manifold X Tx → ∃N : set, ∃e : set, N = m ∪ m ∪ (Sing Empty) ∧ embedding_of X Tx (euclidean_space N) (euclidean_topology N) e ∧ closed_in (euclidean_space N) (euclidean_topology N) (apply_fun e X)
In Proofgold the corresponding term root is 6cd055... and proposition id is c0ff98...
Proof:
Let X, Tx and m be given.
Assume Hm Hman.
We will prove ∃N : set, ∃e : set, N = m ∪ m ∪ (Sing Empty) ∧ embedding_of X Tx (euclidean_space N) (euclidean_topology N) e ∧ closed_in (euclidean_space N) (euclidean_topology N) (apply_fun e X).
The rest of this subproof is missing.
∎
Definition. We define sigma_compact to be λX Tx ⇒ ∃Fam : set, countable Fam ∧ (∀C : set, C ∈ Fam → C ⊆ X ∧ compact_space C (subspace_topology X Tx C)) ∧ X = ⋃ {C ∈ Fam|∃U : set, open_in X Tx U ∧ C ⊆ U} of type set → set → prop.
In Proofgold the corresponding term root is c53cf4... and object id is 044740...
Theorem. (ex50_8_sigma_compact_dimension)
∀X Tx m : set, m ∈ ω → sigma_compact X Tx → Hausdorff_space X Tx → (∀C : set, C ⊆ X → compact_space C (subspace_topology X Tx C) → covering_dimension C m) → covering_dimension X m
In Proofgold the corresponding term root is e2578a... and proposition id is 8a72c7...
Proof:
Let X, Tx and m be given.
Assume Hm Hsig HHaus Hdim.
We will prove covering_dimension X m.
The rest of this subproof is missing.
∎
Theorem. (ex50_9_manifold_dimension_le_m)
∀X Tx m : set, m ∈ ω → m_manifold X Tx → covering_dimension X m
In Proofgold the corresponding term root is edf2d4... and proposition id is f7dd82...
Proof:
Let X, Tx and m be given.
Assume Hm Hman.
The rest of this subproof is missing.
∎
Theorem. (ex50_10_closed_subspace_RN_dimension)
∀X N : set, N ∈ ω → X ⊆ (euclidean_space N) → closed_in (euclidean_space N) (euclidean_topology N) X → covering_dimension X N
In Proofgold the corresponding term root is c9efb3... and proposition id is 84210f...
Proof:
Let X and N be given.
Assume HN Hsub Hclosed.
The rest of this subproof is missing.
∎
Theorem. (ex50_11_embedding_characterization)
∀X Tx : set, (∃N : set, ∃e : set, N ∈ ω ∧ embedding_of X Tx (euclidean_space N) (euclidean_topology N) e ∧ closed_in (euclidean_space N) (euclidean_topology N) (apply_fun e X)) ↔ (locally_compact X Tx ∧ Hausdorff_space X Tx ∧ second_countable_space X Tx ∧ finite_dimensional_space X Tx)
In Proofgold the corresponding term root is f6f0a7... and proposition id is 801659...
Proof:
Let X and Tx be given.
We will prove (∃N : set, ∃e : set, N ∈ ω ∧ embedding_of X Tx (euclidean_space N) (euclidean_topology N) e ∧ closed_in (euclidean_space N) (euclidean_topology N) (apply_fun e X)) ↔ (locally_compact X Tx ∧ Hausdorff_space X Tx ∧ second_countable_space X Tx ∧ finite_dimensional_space X Tx).
The rest of this subproof is missing.
∎
Theorem. (supp_ex_locally_euclidean_1)
∀X Tx m : set, locally_m_euclidean X Tx m → locally_compact X Tx ∧ True
In Proofgold the corresponding term root is a01b1b... and proposition id is db17bb...
Proof:
Let X, Tx and m be given.
Assume Hloc.
The rest of this subproof is missing.
∎
Theorem. (supp_ex_locally_euclidean_2_i_implies_ii)
∀X Tx m : set, locally_m_euclidean X Tx m → compact_space X Tx → Hausdorff_space X Tx → m_manifold X Tx
In Proofgold the corresponding term root is 9f9ebb... and proposition id is fa05f4...
Proof:
Let X, Tx and m be given.
Assume Hloc Hcomp HHaus.
We will prove m_manifold X Tx.
The rest of this subproof is missing.
∎
Theorem. (supp_ex_locally_euclidean_2_ii_implies_iii)
∀X Tx m : set, locally_m_euclidean X Tx m → m_manifold X Tx → metrizable X Tx
In Proofgold the corresponding term root is ac00d2... and proposition id is 39a181...
Proof:
Let X, Tx and m be given.
Assume Hloc Hman.
We will prove metrizable X Tx.
The rest of this subproof is missing.
∎
Theorem. (supp_ex_locally_euclidean_2_iii_implies_iv)
∀X Tx : set, metrizable X Tx → normal_space X Tx
In Proofgold the corresponding term root is 40b24f... and proposition id is f457d9...
Proof:
Let X and Tx be given.
Assume Hmet.
We will prove normal_space X Tx.
The rest of this subproof is missing.
∎
Theorem. (supp_ex_locally_euclidean_2_iv_implies_v)
∀X Tx : set, normal_space X Tx → Hausdorff_space X Tx
In Proofgold the corresponding term root is d76e84... and proposition id is db326f...
Proof:
Let X and Tx be given.
Assume Hnorm.
We will prove Hausdorff_space X Tx.
The rest of this subproof is missing.
∎
Theorem. (supp_ex_locally_euclidean_3)
locally_m_euclidean R R_standard_topology (Sing Empty) ∧ m_manifold R R_standard_topology ∧ ¬ (compact_space R R_standard_topology ∧ Hausdorff_space R R_standard_topology)
In Proofgold the corresponding term root is 5e882e... and proposition id is 37e386...
Theorem. (supp_ex_locally_euclidean_4)
∀Tdict : set, Tdict = R → locally_m_euclidean R Tdict (Sing Empty) ∧ metrizable R Tdict ∧ ¬ m_manifold R Tdict
In Proofgold the corresponding term root is c63891... and proposition id is a5193b...
Proof:
Let Tdict be given.
Assume HTdict.
We will prove locally_m_euclidean R Tdict (Sing Empty) ∧ metrizable R Tdict ∧ ¬ m_manifold R Tdict.
The rest of this subproof is missing.
∎
Theorem. (supp_ex_locally_euclidean_5)
∀L TL : set, L = R → TL = R → locally_m_euclidean L TL (Sing Empty) ∧ normal_space L TL ∧ ¬ metrizable L TL
In Proofgold the corresponding term root is a60239... and proposition id is 35ced5...
Proof:
Let L and TL be given.
Assume HL HTL.
We will prove locally_m_euclidean L TL (Sing Empty) ∧ normal_space L TL ∧ ¬ metrizable L TL.
The rest of this subproof is missing.
∎
Theorem. (supp_ex_locally_euclidean_7)
∀X Tx m : set, locally_m_euclidean X Tx m → (Hausdorff_space X Tx ↔ completely_regular_space X Tx)
In Proofgold the corresponding term root is a49328... and proposition id is 6b26fc...
Proof:
Let X, Tx and m be given.
Assume Hloc.
We will prove Hausdorff_space X Tx ↔ completely_regular_space X Tx.
The rest of this subproof is missing.
∎
Theorem. (supp_ex_locally_euclidean_8)
∀X Tx m : set, locally_m_euclidean X Tx m → (metrizable X Tx ↔ (paracompact_space X Tx ∧ Hausdorff_space X Tx))
In Proofgold the corresponding term root is d4056e... and proposition id is 24def6...
Proof:
Let X, Tx and m be given.
Assume Hloc.
We will prove metrizable X Tx ↔ (paracompact_space X Tx ∧ Hausdorff_space X Tx).
The rest of this subproof is missing.
∎
Theorem. (supp_ex_locally_euclidean_9)
∀X Tx m : set, locally_m_euclidean X Tx m → metrizable X Tx → ∀C : set, C ⊆ X → m_manifold C (subspace_topology X Tx C)
In Proofgold the corresponding term root is b1fdcd... and proposition id is b8107a...
Proof:
Let X, Tx and m be given.
Assume Hloc Hmet.
Let C be given.
Assume HC.
We will prove m_manifold C (subspace_topology X Tx C).
The rest of this subproof is missing.
∎
Definition. We define Gdelta_in to be λX Tx A ⇒ ∃Fam : set, countable Fam ∧ (∀U ∈ Fam, open_in X Tx U) ∧ Intersection_Fam Fam = A of type set → set → set → prop.
In Proofgold the corresponding term root is b3fd1c... and object id is c45053...
Definition. We define open_map to be λX Tx Y Ty f ⇒ topology_on X Tx ∧ topology_on Y Ty ∧ function_on f X Y ∧ ∀U : set, U ∈ Tx → apply_fun f U ∈ Ty of type set → set → set → set → set → prop.
In Proofgold the corresponding term root is 7e3857... and object id is fe7112...
Definition. We define topological_group to be λG Tg ⇒ topology_on G Tg ∧ ∃mult inv e : set, function_on mult (OrderedPair G G) G ∧ function_on inv G G ∧ e ∈ G ∧ continuous_map (OrderedPair G G) (product_topology G Tg G Tg) G Tg mult ∧ continuous_map G Tg G Tg inv of type set → set → prop.
In Proofgold the corresponding term root is 986ab2... and object id is d9c344...
Definition. We define separated_subsets to be λX Tx A B ⇒ closure_of X Tx A ∩ B = Empty ∧ A ∩ closure_of X Tx B = Empty of type set → set → set → set → prop.
In Proofgold the corresponding term root is 26e3a7... and object id is 7afc78...
Definition. We define completely_normal_space to be λX Tx ⇒ normal_space X Tx ∧ (∀A B : set, separated_subsets X Tx A B → ∃U V : set, open_in X Tx U ∧ open_in X Tx V ∧ A ⊆ U ∧ B ⊆ V ∧ U ∩ V = Empty) of type set → set → prop.
In Proofgold the corresponding term root is c2de59... and object id is 9d3e0a...
Definition. We define linear_continuum to be λX Tx ⇒ ∃less : set → set → prop, Tx = order_topology X ∧ (∀A : set, A ⊆ X → A ≠ Empty → (∃upper : set, upper ∈ X ∧ ∀a : set, a ∈ A → less a upper) → ∃lub : set, lub ∈ X ∧ (∀a : set, a ∈ A → less a lub ∨ a = lub) ∧ (∀bound : set, bound ∈ X → (∀a : set, a ∈ A → less a bound ∨ a = bound) → less lub bound ∨ lub = bound)) of type set → set → prop.
In Proofgold the corresponding term root is 5a90ec... and object id is d2c463...
Theorem. (ex30_1a_onepoint_Gdelta_firstcountable_T1)
∀X Tx x : set, first_countable_space X Tx → T1_space X Tx → x ∈ X → Gdelta_in X Tx (Sing x)
In Proofgold the corresponding term root is 59c727... and proposition id is 45c90c...
Proof:
Let X, Tx and x be given.
Assume H1: first_countable_space X Tx.
Assume H2: T1_space X Tx.
Assume H3: x ∈ X.
We will prove Gdelta_in X Tx (Sing x).
The rest of this subproof is missing.
∎
Theorem. (ex30_1b_Gdelta_not_firstcountable_exists)
∃X : set, ∃Tx : set, topology_on X Tx ∧ (∀x : set, x ∈ X → Gdelta_in X Tx (Sing x)) ∧ ¬ first_countable_space X Tx
In Proofgold the corresponding term root is 5dcd7b... and proposition id is 5f68ca...
Proof:
We will prove ∃X : set, ∃Tx : set, topology_on X Tx ∧ (∀x : set, x ∈ X → Gdelta_in X Tx (Sing x)) ∧ ¬ first_countable_space X Tx.
The rest of this subproof is missing.
∎
Theorem. (ex30_2_basis_contains_countable)
∀X Tx : set, ∀Basis : set, second_countable_space X Tx → basis_on X Basis → ∃CountableSub : set, CountableSub ⊆ Basis ∧ countable CountableSub ∧ basis_on X CountableSub
In Proofgold the corresponding term root is d75878... and proposition id is fe10e6...
Proof:
Let X, Tx and Basis be given.
Assume H1: second_countable_space X Tx.
Assume H2: basis_on X Basis.
We will prove ∃CountableSub : set, CountableSub ⊆ Basis ∧ countable CountableSub ∧ basis_on X CountableSub.
The rest of this subproof is missing.
∎
Theorem. (ex30_3_uncountably_many_limit_points)
∀X Tx A : set, second_countable_space X Tx → A ⊆ X → ¬ countable A → ¬ countable {x ∈ A|limit_point_of x A X Tx}
In Proofgold the corresponding term root is b1b117... and proposition id is 4e5242...
Proof:
Let X, Tx and A be given.
Assume H1: second_countable_space X Tx.
Assume H2: A ⊆ X.
Assume H3: ¬ countable A.
We will prove ¬ countable {x ∈ A|limit_point_of x A X Tx}.
The rest of this subproof is missing.
∎
Theorem. (ex30_4_compact_metrizable_second_countable)
∀X Tx d : set, compact_space X Tx → metrizable X Tx → metric_on X d → Tx = metric_topology X d → second_countable_space X Tx
In Proofgold the corresponding term root is fdb7f8... and proposition id is 17da9d...
Proof:
Let X, Tx and d be given.
Assume H1: compact_space X Tx.
Assume H2: metrizable X Tx.
Assume H3: metric_on X d.
Assume H4: Tx = metric_topology X d.
We will prove second_countable_space X Tx.
The rest of this subproof is missing.
∎
Theorem. (ex30_5a_metrizable_countable_dense_second_countable)
∀X Tx : set, metrizable X Tx → (∃D : set, D ⊆ X ∧ countable D ∧ dense_in D X Tx) → second_countable_space X Tx
In Proofgold the corresponding term root is 9ab76c... and proposition id is 3c7893...
Proof:
Let X and Tx be given.
Assume Hmet: metrizable X Tx.
Assume Hdense: ∃D : set, D ⊆ X ∧ countable D ∧ dense_in D X Tx.
We will prove second_countable_space X Tx.
The rest of this subproof is missing.
∎
Theorem. (ex30_5b_metrizable_Lindelof_second_countable)
∀X Tx : set, metrizable X Tx → Lindelof_space X Tx → second_countable_space X Tx
In Proofgold the corresponding term root is 4ba4ca... and proposition id is 926924...
Proof:
Let X and Tx be given.
Assume Hmet: metrizable X Tx.
Assume Hlin: Lindelof_space X Tx.
We will prove second_countable_space X Tx.
The rest of this subproof is missing.
∎
Theorem. (ex30_6a_Rl_not_metrizable)
¬ metrizable R R_lower_limit_topology
In Proofgold the corresponding term root is 8aa8ac... and proposition id is e92c40...
Proof:
We will prove ¬ metrizable R R_lower_limit_topology.
The rest of this subproof is missing.
∎
Theorem. (ex30_6b_ordered_square_not_metrizable)
∀Tx : set, Tx = R → ¬ metrizable ordered_square Tx
In Proofgold the corresponding term root is 0b2b63... and proposition id is 7a7493...
Proof:
Let Tx be given.
Assume H: Tx = R.
We will prove ¬ metrizable ordered_square Tx.
The rest of this subproof is missing.
∎
Theorem. (ex30_7_SOmega_Sbar_Omega_countability)
∀Tx_SO Tx_SbarO : set, Tx_SO = SOmega_topology → Tx_SbarO = SbarOmega_topology → (first_countable_space S_Omega Tx_SO ∧ second_countable_space S_Omega Tx_SO ∧ Lindelof_space S_Omega Tx_SO ∧ (∃D : set, D ⊆ S_Omega ∧ countable D ∧ dense_in D S_Omega Tx_SO)) ∧ (first_countable_space Sbar_Omega Tx_SbarO ∧ ¬ second_countable_space Sbar_Omega Tx_SbarO ∧ ¬ Lindelof_space Sbar_Omega Tx_SbarO ∧ ¬ (∃D : set, D ⊆ Sbar_Omega ∧ countable D ∧ dense_in D Sbar_Omega Tx_SbarO))
In Proofgold the corresponding term root is 6343dd... and proposition id is 2668e5...
Proof:
Let Tx_SO and Tx_SbarO be given.
Assume H1: Tx_SO = SOmega_topology.
Assume H2: Tx_SbarO = SbarOmega_topology.
We will prove (first_countable_space S_Omega Tx_SO ∧ second_countable_space S_Omega Tx_SO ∧ Lindelof_space S_Omega Tx_SO ∧ (∃D : set, D ⊆ S_Omega ∧ countable D ∧ dense_in D S_Omega Tx_SO)) ∧ (first_countable_space Sbar_Omega Tx_SbarO ∧ ¬ second_countable_space Sbar_Omega Tx_SbarO ∧ ¬ Lindelof_space Sbar_Omega Tx_SbarO ∧ ¬ (∃D : set, D ⊆ Sbar_Omega ∧ countable D ∧ dense_in D Sbar_Omega Tx_SbarO)).
The rest of this subproof is missing.
∎
Theorem. (ex30_8_Romega_uniform_countability)
∀Tx : set, Tx = R → first_countable_space R Tx ∧ ¬ second_countable_space R Tx ∧ ¬ Lindelof_space R Tx ∧ ¬ (∃D : set, D ⊆ R ∧ countable D ∧ dense_in D R Tx)
In Proofgold the corresponding term root is 05291c... and proposition id is edf72b...
Proof:
Let Tx be given.
Assume H: Tx = R.
We will prove first_countable_space R Tx ∧ ¬ second_countable_space R Tx ∧ ¬ Lindelof_space R Tx ∧ ¬ (∃D : set, D ⊆ R ∧ countable D ∧ dense_in D R Tx).
The rest of this subproof is missing.
∎
Theorem. (ex30_9a_closed_Lindelof)
∀X Tx A : set, Lindelof_space X Tx → closed_in X Tx A → Lindelof_space A (subspace_topology X Tx A)
In Proofgold the corresponding term root is 462c12... and proposition id is 8b9a5e...
Proof:
Let X, Tx and A be given.
Assume Hlin: Lindelof_space X Tx.
Assume Hcl: closed_in X Tx A.
We will prove Lindelof_space A (subspace_topology X Tx A).
The rest of this subproof is missing.
∎
Theorem. (ex30_9b_dense_not_countable_dense)
∃X : set, ∃Tx : set, ∃A : set, (∃D : set, D ⊆ X ∧ countable D ∧ dense_in D X Tx) ∧ dense_in A X Tx ∧ ¬ (∃DA : set, DA ⊆ A ∧ countable DA ∧ dense_in DA A (subspace_topology X Tx A))
In Proofgold the corresponding term root is 5600a7... and proposition id is fea0ba...
Proof:
We will prove ∃X : set, ∃Tx : set, ∃A : set, (∃D : set, D ⊆ X ∧ countable D ∧ dense_in D X Tx) ∧ dense_in A X Tx ∧ ¬ (∃DA : set, DA ⊆ A ∧ countable DA ∧ dense_in DA A (subspace_topology X Tx A)).
The rest of this subproof is missing.
∎
Theorem. (ex30_10_product_countable_dense)
∀Idx : set, ∀Fam : set, countable Idx → (∀i : set, i ∈ Idx → ∃Xi : set, ∃Txi : set, ∃Di : set, apply_fun Fam i = OrderedPair Xi Txi ∧ Di ⊆ Xi ∧ countable Di ∧ dense_in Di Xi Txi) → ∃D : set, D ⊆ product_space Idx Fam ∧ countable D ∧ dense_in D (product_space Idx Fam) (product_topology_full Idx Fam)
In Proofgold the corresponding term root is f8f7bc... and proposition id is 5912fc...
Proof:
Let Idx and Fam be given.
Assume H1: countable Idx.
Assume H2: ∀i : set, i ∈ Idx → ∃Xi : set, ∃Txi : set, ∃Di : set, apply_fun Fam i = OrderedPair Xi Txi ∧ Di ⊆ Xi ∧ countable Di ∧ dense_in Di Xi Txi.
We will prove ∃D : set, D ⊆ product_space Idx Fam ∧ countable D ∧ dense_in D (product_space Idx Fam) (product_topology_full Idx Fam).
The rest of this subproof is missing.
∎
Theorem. (ex30_11a_image_Lindelof)
∀X Tx Y Ty f : set, Lindelof_space X Tx → continuous_map X Tx Y Ty f → Lindelof_space (apply_fun f X) (subspace_topology Y Ty (apply_fun f X))
In Proofgold the corresponding term root is f5b039... and proposition id is 943462...
Proof:
Let X, Tx, Y, Ty and f be given.
Assume H1: Lindelof_space X Tx.
Assume H2: continuous_map X Tx Y Ty f.
We will prove Lindelof_space (apply_fun f X) (subspace_topology Y Ty (apply_fun f X)).
The rest of this subproof is missing.
∎
Theorem. (ex30_11b_image_countable_dense)
∀X Tx Y Ty f : set, (∃D : set, D ⊆ X ∧ countable D ∧ dense_in D X Tx) → continuous_map X Tx Y Ty f → ∃Df : set, Df ⊆ (apply_fun f X) ∧ countable Df ∧ dense_in Df (apply_fun f X) (subspace_topology Y Ty (apply_fun f X))
In Proofgold the corresponding term root is bbe711... and proposition id is f9c856...
Proof:
Let X, Tx, Y, Ty and f be given.
Assume H1: ∃D : set, D ⊆ X ∧ countable D ∧ dense_in D X Tx.
Assume H2: continuous_map X Tx Y Ty f.
We will prove ∃Df : set, Df ⊆ (apply_fun f X) ∧ countable Df ∧ dense_in Df (apply_fun f X) (subspace_topology Y Ty (apply_fun f X)).
The rest of this subproof is missing.
∎
Theorem. (ex30_12a_open_map_first_countable)
∀X Tx Y Ty f : set, first_countable_space X Tx → continuous_map X Tx Y Ty f → open_map X Tx Y Ty f → first_countable_space (apply_fun f X) (subspace_topology Y Ty (apply_fun f X))
In Proofgold the corresponding term root is a15ff3... and proposition id is 41d402...
Proof:
Let X, Tx, Y, Ty and f be given.
Assume H1: first_countable_space X Tx.
Assume H2: continuous_map X Tx Y Ty f.
Assume H3: open_map X Tx Y Ty f.
We will prove first_countable_space (apply_fun f X) (subspace_topology Y Ty (apply_fun f X)).
The rest of this subproof is missing.
∎
Theorem. (ex30_12b_open_map_second_countable)
∀X Tx Y Ty f : set, second_countable_space X Tx → continuous_map X Tx Y Ty f → open_map X Tx Y Ty f → second_countable_space (apply_fun f X) (subspace_topology Y Ty (apply_fun f X))
In Proofgold the corresponding term root is 9ee628... and proposition id is 33b07e...
Proof:
Let X, Tx, Y, Ty and f be given.
Assume H1: second_countable_space X Tx.
Assume H2: continuous_map X Tx Y Ty f.
Assume H3: open_map X Tx Y Ty f.
We will prove second_countable_space (apply_fun f X) (subspace_topology Y Ty (apply_fun f X)).
The rest of this subproof is missing.
∎
Theorem. (ex30_13_disjoint_open_sets_countable)
∀X Tx : set, (∃D : set, D ⊆ X ∧ countable D ∧ dense_in D X Tx) → ∀Fam : set, (∀U : set, U ∈ Fam → open_in X Tx U) → (∀U V : set, U ∈ Fam → V ∈ Fam → U ≠ V → U ∩ V = Empty) → countable Fam
In Proofgold the corresponding term root is 3b260e... and proposition id is 03dad7...
Proof:
Let X and Tx be given.
Assume H1: ∃D : set, D ⊆ X ∧ countable D ∧ dense_in D X Tx.
Let Fam be given.
Assume H2: ∀U : set, U ∈ Fam → open_in X Tx U.
Assume H3: ∀U V : set, U ∈ Fam → V ∈ Fam → U ≠ V → U ∩ V = Empty.
We will prove countable Fam.
The rest of this subproof is missing.
∎
Theorem. (ex30_14_product_Lindelof_compact)
∀X Tx Y Ty Idx Fam : set, Lindelof_space X Tx → compact_space Y Ty → Lindelof_space (product_space Idx Fam) (product_topology_full Idx Fam)
In Proofgold the corresponding term root is 29a88f... and proposition id is 8858b5...
Proof:
Let X, Tx, Y, Ty, Idx and Fam be given.
Assume H1: Lindelof_space X Tx.
Assume H2: compact_space Y Ty.
We will prove Lindelof_space (product_space Idx Fam) (product_topology_full Idx Fam).
The rest of this subproof is missing.
∎
Theorem. (ex30_15_CI_has_countable_dense_uniform)
∃CI : set, ∃TCI : set, ∃D : set, D ⊆ CI ∧ countable D ∧ dense_in D CI TCI ∧ second_countable_space CI TCI
In Proofgold the corresponding term root is 49c162... and proposition id is ca5f4b...
Proof:
We will prove ∃CI : set, ∃TCI : set, ∃D : set, D ⊆ CI ∧ countable D ∧ dense_in D CI TCI ∧ second_countable_space CI TCI.
The rest of this subproof is missing.
∎
Theorem. (ex30_16a_product_RI_countable_dense)
∃Idx : set, ∃Fam : set, ∃D : set, D ⊆ product_space Idx Fam ∧ countable D ∧ dense_in D (product_space Idx Fam) (product_topology_full Idx Fam)
In Proofgold the corresponding term root is d7211a... and proposition id is 670d33...
Proof:
We will prove ∃Idx : set, ∃Fam : set, ∃D : set, D ⊆ product_space Idx Fam ∧ countable D ∧ dense_in D (product_space Idx Fam) (product_topology_full Idx Fam).
The rest of this subproof is missing.
∎
Theorem. (ex30_16b_large_product_no_countable_dense)
∀J : set, atleastp (𝒫 ω) J → ¬ equip J (𝒫 ω) → ∀Fam : set, ¬ (∃D : set, D ⊆ product_space J Fam ∧ countable D ∧ dense_in D (product_space J Fam) (product_topology_full J Fam))
In Proofgold the corresponding term root is edf415... and proposition id is fdf3ab...
Proof:
Let J be given.
Assume H1: atleastp (𝒫 ω) J.
Assume H2: ¬ equip J (𝒫 ω).
Let Fam be given.
We will prove ¬ (∃D : set, D ⊆ product_space J Fam ∧ countable D ∧ dense_in D (product_space J Fam) (product_topology_full J Fam)).
The rest of this subproof is missing.
∎
Theorem. (ex30_17_Romega_box_countability)
∃Romega : set, ∃BoxTop : set, ∃Qinf : set, ∃SubTop : set, SubTop = subspace_topology Romega BoxTop Qinf ∧ (first_countable_space Qinf SubTop ∨ ¬ first_countable_space Qinf SubTop) ∧ (second_countable_space Qinf SubTop ∨ ¬ second_countable_space Qinf SubTop) ∧ (Lindelof_space Qinf SubTop ∨ ¬ Lindelof_space Qinf SubTop) ∧ ((∃D : set, D ⊆ Qinf ∧ countable D ∧ dense_in D Qinf SubTop) ∨ ¬ (∃D : set, D ⊆ Qinf ∧ countable D ∧ dense_in D Qinf SubTop))
In Proofgold the corresponding term root is 4e5d50... and proposition id is 07858e...
Proof:
We will prove ∃Romega : set, ∃BoxTop : set, ∃Qinf : set, ∃SubTop : set, SubTop = subspace_topology Romega BoxTop Qinf ∧ (first_countable_space Qinf SubTop ∨ ¬ first_countable_space Qinf SubTop) ∧ (second_countable_space Qinf SubTop ∨ ¬ second_countable_space Qinf SubTop) ∧ (Lindelof_space Qinf SubTop ∨ ¬ Lindelof_space Qinf SubTop) ∧ ((∃D : set, D ⊆ Qinf ∧ countable D ∧ dense_in D Qinf SubTop) ∨ ¬ (∃D : set, D ⊆ Qinf ∧ countable D ∧ dense_in D Qinf SubTop)).
The rest of this subproof is missing.
∎
Theorem. (ex30_18_first_countable_group_countable_basis)
∀G Tg : set, topological_group G Tg → first_countable_space G Tg → ((∃D : set, D ⊆ G ∧ countable D ∧ dense_in D G Tg) ∨ Lindelof_space G Tg) → second_countable_space G Tg
In Proofgold the corresponding term root is 416892... and proposition id is b7835e...
Proof:
Let G and Tg be given.
Assume H1: topological_group G Tg.
Assume H2: first_countable_space G Tg.
Assume H3: (∃D : set, D ⊆ G ∧ countable D ∧ dense_in D G Tg) ∨ Lindelof_space G Tg.
We will prove second_countable_space G Tg.
The rest of this subproof is missing.
∎
Theorem. (ex31_1_regular_disjoint_closure_neighborhoods)
∀X Tx x y : set, regular_space X Tx → x ∈ X → y ∈ X → x ≠ y → ∃U V : set, open_in X Tx U ∧ open_in X Tx V ∧ x ∈ U ∧ y ∈ V ∧ closure_of X Tx U ∩ closure_of X Tx V = Empty
In Proofgold the corresponding term root is c64b78... and proposition id is dcf57c...
Proof:
Let X, Tx, x and y be given.
Assume Hreg: regular_space X Tx.
Assume Hx: x ∈ X.
Assume Hy: y ∈ X.
Assume Hneq: x ≠ y.
We will prove ∃U V : set, open_in X Tx U ∧ open_in X Tx V ∧ x ∈ U ∧ y ∈ V ∧ closure_of X Tx U ∩ closure_of X Tx V = Empty.
The rest of this subproof is missing.
∎
Theorem. (ex31_2_normal_disjoint_closure_neighborhoods)
∀X Tx A B : set, normal_space X Tx → closed_in X Tx A → closed_in X Tx B → A ∩ B = Empty → ∃U V : set, open_in X Tx U ∧ open_in X Tx V ∧ A ⊆ U ∧ B ⊆ V ∧ closure_of X Tx U ∩ closure_of X Tx V = Empty
In Proofgold the corresponding term root is 528ce9... and proposition id is 6c5039...
Proof:
Let X, Tx, A and B be given.
Assume Hnorm: normal_space X Tx.
Assume HA: closed_in X Tx A.
Assume HB: closed_in X Tx B.
Assume Hdisj: A ∩ B = Empty.
We will prove ∃U V : set, open_in X Tx U ∧ open_in X Tx V ∧ A ⊆ U ∧ B ⊆ V ∧ closure_of X Tx U ∩ closure_of X Tx V = Empty.
The rest of this subproof is missing.
∎
Theorem. (ex31_3_order_topology_regular)
∀X : set, regular_space X (order_topology X)
In Proofgold the corresponding term root is 4cd9dc... and proposition id is 461426...
Proof:
Let X be given.
We will prove regular_space X (order_topology X).
The rest of this subproof is missing.
∎
Theorem. (ex31_4_comparison_topologies_separation)
∀X Tx Tx' : set, Tx ⊆ Tx' → ((Hausdorff_space X Tx' → Hausdorff_space X Tx) ∧ (regular_space X Tx → regular_space X Tx') ∧ (normal_space X Tx → normal_space X Tx'))
In Proofgold the corresponding term root is 51abb5... and proposition id is c1fe25...
Proof:
Let X, Tx and Tx' be given.
Assume Hfiner: Tx ⊆ Tx'.
Apply and3I to the current goal.
We will prove Hausdorff_space X Tx' → Hausdorff_space X Tx.
The rest of this subproof is missing.
We will prove regular_space X Tx → regular_space X Tx'.
The rest of this subproof is missing.
We will prove normal_space X Tx → normal_space X Tx'.
The rest of this subproof is missing.
∎
Theorem. (ex31_5_equalizer_closed_in_Hausdorff)
∀X Tx Y Ty f g : set, continuous_map X Tx Y Ty f → continuous_map X Tx Y Ty g → Hausdorff_space Y Ty → closed_in X Tx {x ∈ X|apply_fun f x = apply_fun g x}
In Proofgold the corresponding term root is 5acdad... and proposition id is b05e3c...
Proof:
Let X, Tx, Y, Ty, f and g be given.
Assume Hf: continuous_map X Tx Y Ty f.
Assume Hg: continuous_map X Tx Y Ty g.
Assume HHaus: Hausdorff_space Y Ty.
We will prove closed_in X Tx {x ∈ X|apply_fun f x = apply_fun g x}.
The rest of this subproof is missing.
∎
Theorem. (ex31_6_closed_map_preserves_normal)
∀X Tx Y Ty p : set, normal_space X Tx → continuous_map X Tx Y Ty p → (∀A : set, closed_in X Tx A → closed_in Y Ty (apply_fun p A)) → (∀y : set, y ∈ Y → ∃x : set, x ∈ X ∧ apply_fun p x = y) → normal_space Y Ty
In Proofgold the corresponding term root is ade8b0... and proposition id is ad12a8...
Proof:
Let X, Tx, Y, Ty and p be given.
Assume Hnorm: normal_space X Tx.
Assume Hcont: continuous_map X Tx Y Ty p.
Assume Hclosed: ∀A : set, closed_in X Tx A → closed_in Y Ty (apply_fun p A).
Assume Hsurj: ∀y : set, y ∈ Y → ∃x : set, x ∈ X ∧ apply_fun p x = y.
We will prove normal_space Y Ty.
The rest of this subproof is missing.
∎
Theorem. (ex31_7_perfect_map_properties)
∀X Tx Y Ty p : set, continuous_map X Tx Y Ty p → (∀A : set, closed_in X Tx A → closed_in Y Ty (apply_fun p A)) → (∀y : set, y ∈ Y → ∃x : set, x ∈ X ∧ apply_fun p x = y) → (∀y : set, y ∈ Y → compact_space {x ∈ X|apply_fun p x = y} (subspace_topology X Tx {x ∈ X|apply_fun p x = y})) → (Hausdorff_space X Tx → Hausdorff_space Y Ty) ∧ (regular_space X Tx → regular_space Y Ty) ∧ (locally_compact X Tx → locally_compact Y Ty) ∧ (second_countable_space X Tx → second_countable_space Y Ty)
In Proofgold the corresponding term root is baa0d0... and proposition id is 0971d6...
Proof:
Let X, Tx, Y, Ty and p be given.
Assume Hcont: continuous_map X Tx Y Ty p.
Assume Hclosed: ∀A : set, closed_in X Tx A → closed_in Y Ty (apply_fun p A).
Assume Hsurj: ∀y : set, y ∈ Y → ∃x : set, x ∈ X ∧ apply_fun p x = y.
Assume Hcompact: ∀y : set, y ∈ Y → compact_space {x ∈ X|apply_fun p x = y} (subspace_topology X Tx {x ∈ X|apply_fun p x = y}).
We will prove (Hausdorff_space X Tx → Hausdorff_space Y Ty) ∧ (regular_space X Tx → regular_space Y Ty) ∧ (locally_compact X Tx → locally_compact Y Ty) ∧ (second_countable_space X Tx → second_countable_space Y Ty).
The rest of this subproof is missing.
∎
Theorem. (ex31_8_orbit_space_properties)
∀G Tg X Tx alpha : set, topological_group G Tg → compact_space G Tg → (Hausdorff_space X Tx → ∃XG TxG : set, Hausdorff_space XG TxG) ∧ (regular_space X Tx → ∃XG TxG : set, regular_space XG TxG) ∧ (normal_space X Tx → ∃XG TxG : set, normal_space XG TxG) ∧ (locally_compact X Tx → ∃XG TxG : set, locally_compact XG TxG) ∧ (second_countable_space X Tx → ∃XG TxG : set, second_countable_space XG TxG)
In Proofgold the corresponding term root is 7487d0... and proposition id is f29309...
Proof:
Let G, Tg, X, Tx and alpha be given.
Assume Hgrp: topological_group G Tg.
Assume Hcomp: compact_space G Tg.
We will prove (Hausdorff_space X Tx → ∃XG TxG : set, Hausdorff_space XG TxG) ∧ (regular_space X Tx → ∃XG TxG : set, regular_space XG TxG) ∧ (normal_space X Tx → ∃XG TxG : set, normal_space XG TxG) ∧ (locally_compact X Tx → ∃XG TxG : set, locally_compact XG TxG) ∧ (second_countable_space X Tx → ∃XG TxG : set, second_countable_space XG TxG).
The rest of this subproof is missing.
∎
Theorem. (ex31_9_Sorgenfrey_plane_no_separation)
∃Rl2 Tl2 A B : set, ¬ (∃U V : set, open_in Rl2 Tl2 U ∧ open_in Rl2 Tl2 V ∧ A ⊆ U ∧ B ⊆ V ∧ U ∩ V = Empty)
In Proofgold the corresponding term root is 530199... and proposition id is d0b597...
Proof:
We will prove ∃Rl2 Tl2 A B : set, ¬ (∃U V : set, open_in Rl2 Tl2 U ∧ open_in Rl2 Tl2 V ∧ A ⊆ U ∧ B ⊆ V ∧ U ∩ V = Empty).
The rest of this subproof is missing.
∎
Theorem. (ex32_1_closed_subspace_normal)
∀X Tx A : set, normal_space X Tx → closed_in X Tx A → normal_space A (subspace_topology X Tx A)
In Proofgold the corresponding term root is 1f6292... and proposition id is 9a751c...
Proof:
Let X, Tx and A be given.
Assume Hnorm: normal_space X Tx.
Assume Hcl: closed_in X Tx A.
We will prove normal_space A (subspace_topology X Tx A).
The rest of this subproof is missing.
∎
Theorem. (ex32_2_factors_inherit_separation)
∀Idx Fam : set, (∀i : set, i ∈ Idx → ∃Xi Txi : set, apply_fun Fam i = OrderedPair Xi Txi ∧ Xi ≠ Empty) → ((Hausdorff_space (product_space Idx Fam) (product_topology_full Idx Fam) → ∀i : set, i ∈ Idx → ∃Xi Txi : set, apply_fun Fam i = OrderedPair Xi Txi ∧ Hausdorff_space Xi Txi) ∧ (regular_space (product_space Idx Fam) (product_topology_full Idx Fam) → ∀i : set, i ∈ Idx → ∃Xi Txi : set, apply_fun Fam i = OrderedPair Xi Txi ∧ regular_space Xi Txi) ∧ (normal_space (product_space Idx Fam) (product_topology_full Idx Fam) → ∀i : set, i ∈ Idx → ∃Xi Txi : set, apply_fun Fam i = OrderedPair Xi Txi ∧ normal_space Xi Txi))
In Proofgold the corresponding term root is cda20b... and proposition id is 73bce7...
Proof:
Let Idx and Fam be given.
Assume Hnemp: ∀i : set, i ∈ Idx → ∃Xi Txi : set, apply_fun Fam i = OrderedPair Xi Txi ∧ Xi ≠ Empty.
We will prove (Hausdorff_space (product_space Idx Fam) (product_topology_full Idx Fam) → ∀i : set, i ∈ Idx → ∃Xi Txi : set, apply_fun Fam i = OrderedPair Xi Txi ∧ Hausdorff_space Xi Txi) ∧ (regular_space (product_space Idx Fam) (product_topology_full Idx Fam) → ∀i : set, i ∈ Idx → ∃Xi Txi : set, apply_fun Fam i = OrderedPair Xi Txi ∧ regular_space Xi Txi) ∧ (normal_space (product_space Idx Fam) (product_topology_full Idx Fam) → ∀i : set, i ∈ Idx → ∃Xi Txi : set, apply_fun Fam i = OrderedPair Xi Txi ∧ normal_space Xi Txi).
The rest of this subproof is missing.
∎
Theorem. (ex32_3_locally_compact_Hausdorff_regular)
∀X Tx : set, locally_compact X Tx → Hausdorff_space X Tx → regular_space X Tx
In Proofgold the corresponding term root is c106f4... and proposition id is 4fda0c...
Proof:
Let X and Tx be given.
Assume Hlc: locally_compact X Tx.
Assume Hh: Hausdorff_space X Tx.
We will prove regular_space X Tx.
The rest of this subproof is missing.
∎
Theorem. (ex32_4_regular_Lindelof_normal)
∀X Tx : set, regular_space X Tx → Lindelof_space X Tx → normal_space X Tx
In Proofgold the corresponding term root is 9d2cda... and proposition id is 6d5522...
Proof:
Let X and Tx be given.
Assume Hreg: regular_space X Tx.
Assume Hlin: Lindelof_space X Tx.
We will prove normal_space X Tx.
The rest of this subproof is missing.
∎
Theorem. (ex32_5_Romega_normality_questions)
(normal_space (product_space ω (const_family ω R)) (product_topology_full ω (const_family ω R)) ∨ ¬ normal_space (product_space ω (const_family ω R)) (product_topology_full ω (const_family ω R))) ∧ (∃Romega Tunif : set, (normal_space Romega Tunif ∨ ¬ normal_space Romega Tunif))
In Proofgold the corresponding term root is f4e40f... and proposition id is c3e4eb...
Proof:
We will prove (normal_space (product_space ω (const_family ω R)) (product_topology_full ω (const_family ω R)) ∨ ¬ normal_space (product_space ω (const_family ω R)) (product_topology_full ω (const_family ω R))) ∧ (∃Romega Tunif : set, (normal_space Romega Tunif ∨ ¬ normal_space Romega Tunif)).
The rest of this subproof is missing.
∎
Theorem. (ex32_6_completely_normal_characterization)
∀X Tx : set, completely_normal_space X Tx ↔ (∀A B : set, separated_subsets X Tx A B → ∃U V : set, open_in X Tx U ∧ open_in X Tx V ∧ A ⊆ U ∧ B ⊆ V ∧ U ∩ V = Empty)
In Proofgold the corresponding term root is 556217... and proposition id is b8a274...
Proof:
Let X and Tx be given.
Apply iffI to the current goal.
Assume H1: completely_normal_space X Tx.
Let A and B be given.
Assume H2: separated_subsets X Tx A B.
Apply H1 to the current goal.
Assume Hnorm Hsep.
An exact proof term for the current goal is Hsep A B H2.
Assume H1: ∀A B : set, separated_subsets X Tx A B → ∃U V : set, open_in X Tx U ∧ open_in X Tx V ∧ A ⊆ U ∧ B ⊆ V ∧ U ∩ V = Empty.
We will prove completely_normal_space X Tx.
We will prove normal_space X Tx ∧ (∀A B : set, separated_subsets X Tx A B → ∃U V : set, open_in X Tx U ∧ open_in X Tx V ∧ A ⊆ U ∧ B ⊆ V ∧ U ∩ V = Empty).
Apply andI to the current goal.
We will prove normal_space X Tx.
The rest of this subproof is missing.
We will prove ∀A B : set, separated_subsets X Tx A B → ∃U V : set, open_in X Tx U ∧ open_in X Tx V ∧ A ⊆ U ∧ B ⊆ V ∧ U ∩ V = Empty.
An exact proof term for the current goal is H1.
∎
Theorem. (ex32_7_completely_normal_examples)
(∀X Tx A : set, completely_normal_space X Tx → completely_normal_space A (subspace_topology X Tx A)) ∧ (∀X Tx Y Ty Idx Fam : set, completely_normal_space X Tx → completely_normal_space Y Ty → (completely_normal_space (product_space Idx Fam) (product_topology_full Idx Fam) ∨ ¬ completely_normal_space (product_space Idx Fam) (product_topology_full Idx Fam))) ∧ (∀X : set, completely_normal_space X (order_topology X)) ∧ (∀X Tx : set, metrizable X Tx → completely_normal_space X Tx) ∧ (∀X Tx : set, compact_space X Tx → Hausdorff_space X Tx → (completely_normal_space X Tx ∨ ¬ completely_normal_space X Tx)) ∧ (∀X Tx : set, regular_space X Tx → second_countable_space X Tx → (completely_normal_space X Tx ∨ ¬ completely_normal_space X Tx)) ∧ (∃Rl Tl : set, completely_normal_space Rl Tl ∨ ¬ completely_normal_space Rl Tl)
In Proofgold the corresponding term root is 8aeabf... and proposition id is d23c6f...
Proof:
We will prove (∀X Tx A : set, completely_normal_space X Tx → completely_normal_space A (subspace_topology X Tx A)) ∧ (∀X Tx Y Ty Idx Fam : set, completely_normal_space X Tx → completely_normal_space Y Ty → (completely_normal_space (product_space Idx Fam) (product_topology_full Idx Fam) ∨ ¬ completely_normal_space (product_space Idx Fam) (product_topology_full Idx Fam))) ∧ (∀X : set, completely_normal_space X (order_topology X)) ∧ (∀X Tx : set, metrizable X Tx → completely_normal_space X Tx) ∧ (∀X Tx : set, compact_space X Tx → Hausdorff_space X Tx → (completely_normal_space X Tx ∨ ¬ completely_normal_space X Tx)) ∧ (∀X Tx : set, regular_space X Tx → second_countable_space X Tx → (completely_normal_space X Tx ∨ ¬ completely_normal_space X Tx)) ∧ (∃Rl Tl : set, completely_normal_space Rl Tl ∨ ¬ completely_normal_space Rl Tl).
The rest of this subproof is missing.
∎
Theorem. (ex32_8_linear_continuum_normal)
∀X Tx : set, linear_continuum X Tx → normal_space X Tx
In Proofgold the corresponding term root is 577435... and proposition id is 2e702b...
Proof:
Let X and Tx be given.
Assume Hlc: linear_continuum X Tx.
We will prove normal_space X Tx.
The rest of this subproof is missing.
∎
Theorem. (ex32_9_uncountable_product_not_normal)
∀J : set, ¬ countable J → ¬ normal_space (product_space J (const_family J R)) (product_topology_full J (const_family J R))
In Proofgold the corresponding term root is da41a7... and proposition id is ea8a11...
Proof:
Let J be given.
Assume Huncnt: ¬ countable J.
We will prove ¬ normal_space (product_space J (const_family J R)) (product_topology_full J (const_family J R)).
The rest of this subproof is missing.
∎
Definition. We define perfectly_normal_space to be λX Tx ⇒ normal_space X Tx ∧ (∀A : set, closed_in X Tx A → Gdelta_in X Tx A) of type set → set → prop.
In Proofgold the corresponding term root is 08c1e5... and object id is 1a7773...
Theorem. (ex33_1_level_sets_urysohn)
∀X Tx A B : set, ∀U : set → set, normal_space X Tx → closed_in X Tx A → closed_in X Tx B → A ∩ B = Empty → ∃f : set, continuous_map X Tx R R_standard_topology f ∧ (∀x : set, x ∈ A → apply_fun f x = 0) ∧ (∀x : set, x ∈ B → apply_fun f x = 1) ∧ (∀r : set, apply_fun f X = {x ∈ X|apply_fun f x = r})
In Proofgold the corresponding term root is 274539... and proposition id is 48175e...
Proof:
Let X, Tx, A, B and U be given.
Assume Hnorm: normal_space X Tx.
Assume HA: closed_in X Tx A.
Assume HB: closed_in X Tx B.
Assume Hdisj: A ∩ B = Empty.
We will prove ∃f : set, continuous_map X Tx R R_standard_topology f ∧ (∀x : set, x ∈ A → apply_fun f x = 0) ∧ (∀x : set, x ∈ B → apply_fun f x = 1) ∧ (∀r : set, apply_fun f X = {x ∈ X|apply_fun f x = r}).
The rest of this subproof is missing.
∎
Theorem. (ex33_2a_connected_normal_uncountable)
∀X Tx : set, connected_space X Tx → normal_space X Tx → (∃x y : set, x ∈ X ∧ y ∈ X ∧ x ≠ y) → ¬ countable X
In Proofgold the corresponding term root is bc0d49... and proposition id is 74be64...
Proof:
Let X and Tx be given.
Assume Hconn: connected_space X Tx.
Assume Hnorm: normal_space X Tx.
Assume Hneq: ∃x y : set, x ∈ X ∧ y ∈ X ∧ x ≠ y.
We will prove ¬ countable X.
The rest of this subproof is missing.
∎
Theorem. (ex33_2b_connected_regular_uncountable)
∀X Tx : set, connected_space X Tx → regular_space X Tx → (∃x y : set, x ∈ X ∧ y ∈ X ∧ x ≠ y) → ¬ countable X
In Proofgold the corresponding term root is e2b05f... and proposition id is 6d60b2...
Proof:
Let X and Tx be given.
Assume Hconn: connected_space X Tx.
Assume Hreg: regular_space X Tx.
Assume Hneq: ∃x y : set, x ∈ X ∧ y ∈ X ∧ x ≠ y.
We will prove ¬ countable X.
The rest of this subproof is missing.
∎
Theorem. (ex33_3_urysohn_metric_direct)
∀X d A B : set, metric_on X d → closed_in X (metric_topology X d) A → closed_in X (metric_topology X d) B → A ∩ B = Empty → ∃f : set, continuous_map X (metric_topology X d) R R_standard_topology f ∧ (∀x : set, x ∈ A → apply_fun f x = 0) ∧ (∀x : set, x ∈ B → apply_fun f x = 1)
In Proofgold the corresponding term root is e7df57... and proposition id is abecb3...
Proof:
Let X, d, A and B be given.
Assume Hmet: metric_on X d.
Assume HcA: closed_in X (metric_topology X d) A.
Assume HcB: closed_in X (metric_topology X d) B.
Assume Hdisj: A ∩ B = Empty.
We will prove ∃f : set, continuous_map X (metric_topology X d) R R_standard_topology f ∧ (∀x : set, x ∈ A → apply_fun f x = 0) ∧ (∀x : set, x ∈ B → apply_fun f x = 1).
The rest of this subproof is missing.
∎
Theorem. (ex33_4_closed_Gdelta_vanishing_function)
∀X Tx A : set, normal_space X Tx → closed_in X Tx A → (Gdelta_in X Tx A ↔ ∃f : set, continuous_map X Tx R R_standard_topology f ∧ (∀x : set, x ∈ A → apply_fun f x = 0) ∧ (∀x : set, x ∉ A → ¬ (apply_fun f x = 0)))
In Proofgold the corresponding term root is 694c43... and proposition id is 2905b6...
Proof:
Let X, Tx and A be given.
Assume Hnorm: normal_space X Tx.
Assume HA: closed_in X Tx A.
Apply iffI to the current goal.
Assume HG: Gdelta_in X Tx A.
We will prove ∃f : set, continuous_map X Tx R R_standard_topology f ∧ (∀x : set, x ∈ A → apply_fun f x = 0) ∧ (∀x : set, x ∉ A → ¬ (apply_fun f x = 0)).
The rest of this subproof is missing.
Assume Hf: ∃f : set, continuous_map X Tx R R_standard_topology f ∧ (∀x : set, x ∈ A → apply_fun f x = 0) ∧ (∀x : set, x ∉ A → ¬ (apply_fun f x = 0)).
We will prove Gdelta_in X Tx A.
The rest of this subproof is missing.
∎
Theorem. (ex33_5_strong_urysohn)
∀X Tx A B : set, normal_space X Tx → closed_in X Tx A → closed_in X Tx B → A ∩ B = Empty → (Gdelta_in X Tx A ∧ Gdelta_in X Tx B ↔ ∃f : set, continuous_map X Tx R R_standard_topology f ∧ (∀x : set, x ∈ A → apply_fun f x = 0) ∧ (∀x : set, x ∈ B → apply_fun f x = 1) ∧ (∀x : set, x ∈ X → x ∉ A → x ∉ B → ¬ (apply_fun f x = 0) ∧ ¬ (apply_fun f x = 1)))
In Proofgold the corresponding term root is e1ace7... and proposition id is 7e195f...
Proof:
Let X, Tx, A and B be given.
Assume Hnorm: normal_space X Tx.
Assume HA: closed_in X Tx A.
Assume HB: closed_in X Tx B.
Assume Hdisj: A ∩ B = Empty.
Apply iffI to the current goal.
Assume HG: Gdelta_in X Tx A ∧ Gdelta_in X Tx B.
We will prove ∃f : set, continuous_map X Tx R R_standard_topology f ∧ (∀x : set, x ∈ A → apply_fun f x = 0) ∧ (∀x : set, x ∈ B → apply_fun f x = 1) ∧ (∀x : set, x ∈ X → x ∉ A → x ∉ B → ¬ (apply_fun f x = 0) ∧ ¬ (apply_fun f x = 1)).
The rest of this subproof is missing.
Assume Hf: ∃f : set, continuous_map X Tx R R_standard_topology f ∧ (∀x : set, x ∈ A → apply_fun f x = 0) ∧ (∀x : set, x ∈ B → apply_fun f x = 1) ∧ (∀x : set, x ∈ X → x ∉ A → x ∉ B → ¬ (apply_fun f x = 0) ∧ ¬ (apply_fun f x = 1)).
We will prove Gdelta_in X Tx A ∧ Gdelta_in X Tx B.
The rest of this subproof is missing.
∎
Theorem. (ex33_6a_metrizable_perfectly_normal)
∀X Tx : set, metrizable X Tx → perfectly_normal_space X Tx
In Proofgold the corresponding term root is 4075ab... and proposition id is 8a88f0...
Proof:
Let X and Tx be given.
Assume Hmet: metrizable X Tx.
We will prove perfectly_normal_space X Tx.
The rest of this subproof is missing.
∎
Theorem. (ex33_6b_perfectly_completely_normal)
∀X Tx : set, perfectly_normal_space X Tx → completely_normal_space X Tx
In Proofgold the corresponding term root is 6435da... and proposition id is 6ceb4c...
Proof:
Let X and Tx be given.
Assume Hperf: perfectly_normal_space X Tx.
We will prove completely_normal_space X Tx.
The rest of this subproof is missing.
∎
Theorem. (ex33_6c_completely_not_perfectly_normal)
∃X Tx : set, completely_normal_space X Tx ∧ ¬ perfectly_normal_space X Tx
In Proofgold the corresponding term root is d0fb84... and proposition id is 3a1a10...
Proof:
We will prove ∃X Tx : set, completely_normal_space X Tx ∧ ¬ perfectly_normal_space X Tx.
The rest of this subproof is missing.
∎
Theorem. (ex33_7_locally_compact_Hausdorff_completely_regular)
∀X Tx : set, locally_compact X Tx → Hausdorff_space X Tx → completely_regular_space X Tx
In Proofgold the corresponding term root is 035e9e... and proposition id is ad4657...
Proof:
Let X and Tx be given.
Assume Hlc: locally_compact X Tx.
Assume Hh: Hausdorff_space X Tx.
We will prove completely_regular_space X Tx.
The rest of this subproof is missing.
∎
Theorem. (ex33_8_compact_subset_continuous_separation)
∀X Tx A B : set, completely_regular_space X Tx → compact_space A (subspace_topology X Tx A) → closed_in X Tx B → A ∩ B = Empty → ∃f : set, continuous_map X Tx R R_standard_topology f ∧ (∀x : set, x ∈ A → apply_fun f x = 0) ∧ (∀x : set, x ∈ B → apply_fun f x = 1)
In Proofgold the corresponding term root is 3c3617... and proposition id is 88874f...
Proof:
Let X, Tx, A and B be given.
Assume Hcr: completely_regular_space X Tx.
Assume Hcpt: compact_space A (subspace_topology X Tx A).
Assume HcB: closed_in X Tx B.
Assume Hdisj: A ∩ B = Empty.
We will prove ∃f : set, continuous_map X Tx R R_standard_topology f ∧ (∀x : set, x ∈ A → apply_fun f x = 0) ∧ (∀x : set, x ∈ B → apply_fun f x = 1).
The rest of this subproof is missing.
∎
Theorem. (ex33_9_Romega_box_completely_regular)
completely_regular_space (product_space ω (const_family ω R)) (box_topology ω (const_family ω R))
In Proofgold the corresponding term root is 63c458... and proposition id is a6f317...
Proof:
We will prove completely_regular_space (product_space ω (const_family ω R)) (box_topology ω (const_family ω R)).
The rest of this subproof is missing.
∎
Theorem. (ex33_10_topological_group_completely_regular)
∀G Tg : set, topological_group G Tg → completely_regular_space G Tg
In Proofgold the corresponding term root is 1e33ef... and proposition id is f0bb84...
Proof:
Let G and Tg be given.
Assume Htg: topological_group G Tg.
We will prove completely_regular_space G Tg.
The rest of this subproof is missing.
∎
Theorem. (ex33_11_regular_not_completely_regular)
∃X Tx : set, regular_space X Tx ∧ ¬ completely_regular_space X Tx
In Proofgold the corresponding term root is 063029... and proposition id is 8e0dba...
Proof:
We will prove ∃X Tx : set, regular_space X Tx ∧ ¬ completely_regular_space X Tx.
The rest of this subproof is missing.
∎
Definition. We define locally_metrizable_space to be λX Tx ⇒ topology_on X Tx ∧ ∀x : set, x ∈ X → ∃N : set, N ∈ Tx ∧ x ∈ N ∧ ∃d : set, metric_on N d ∧ subspace_topology X Tx N = metric_topology N d of type set → set → prop.
In Proofgold the corresponding term root is 181d4a... and object id is 3dbbe8...
Definition. We define retraction_of to be λX Tx A ⇒ A ⊆ X ∧ ∃r : set, function_on r X X ∧ continuous_map X Tx X Tx r ∧ (∀x : set, x ∈ X → apply_fun r x ∈ A) ∧ (∀x : set, x ∈ A → apply_fun r x = x) of type set → set → set → prop.
In Proofgold the corresponding term root is 649a5f... and object id is d0e1e2...
Definition. We define image_of_map to be λX Tx Y Ty f ⇒ {apply_fun f x|x ∈ X} of type set → set → set → set → set → set.
In Proofgold the corresponding term root is 4ab3c7... and object id is 1acaf0...
Definition. We define absolute_retract to be λX Tx ⇒ Hausdorff_space X Tx ∧ ∀Y Ty, normal_space Y Ty → ∃e : set, embedding_of X Tx Y Ty e ∧ ∃r : set, retraction_of Y Ty (image_of_map X Tx Y Ty e) of type set → set → prop.
In Proofgold the corresponding term root is a85b45... and object id is 271cfd...
Definition. We define coherent_topology to be λX Tx Y Ty ⇒ topology_on X Tx ∧ topology_on Y Ty ∧ X ⊆ Y ∧ subspace_topology Y Ty X = Tx of type set → set → set → set → prop.
In Proofgold the corresponding term root is c53716... and object id is 3240c7...
Definition. We define compact_spaces_family to be λI Xi ⇒ ∀i : set, i ∈ I → compact_space (product_component Xi i) (product_component_topology Xi i) of type set → set → prop.
In Proofgold the corresponding term root is 3d618e... and object id is d22e23...
Definition. We define surjective_map to be λX Y f ⇒ function_on f X Y ∧ ∀y : set, y ∈ Y → ∃x : set, x ∈ X ∧ apply_fun f x = y of type set → set → set → prop.
In Proofgold the corresponding term root is c10c8d... and object id is 45c8ee...
Definition. We define ex34_1_Hausdorff_countable_basis_not_metrizable_example to be {p ∈ 𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx : set, p = OrderedPair X Tx ∧ Hausdorff_space X Tx ∧ second_countable_space X Tx ∧ ¬ metrizable X Tx} of type set.
In Proofgold the corresponding term root is 9ed9fb... and object id is 128e56...
Definition. We define ex34_2_completely_normal_not_metrizable_example to be {p ∈ 𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx : set, p = OrderedPair X Tx ∧ completely_normal_space X Tx ∧ ¬ metrizable X Tx} of type set.
In Proofgold the corresponding term root is 76e022... and object id is 685168...
Definition. We define ex34_3_compact_Hausdorff_metrizable_iff_second_countable to be {p ∈ 𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx : set, p = OrderedPair X Tx ∧ compact_space X Tx ∧ Hausdorff_space X Tx ∧ (metrizable X Tx ↔ second_countable_space X Tx)} of type set.
In Proofgold the corresponding term root is ac9d4d... and object id is 18d631...
Definition. We define ex34_4_locally_compact_Hausdorff_metrizable_questions to be {p ∈ 𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx : set, p = OrderedPair X Tx ∧ locally_compact X Tx ∧ Hausdorff_space X Tx ∧ (second_countable_space X Tx → metrizable X Tx)} of type set.
In Proofgold the corresponding term root is 0b45f1... and object id is 2938b3...
Definition. We define ex34_5_one_point_compactification_metrizable_questions to be {q ∈ 𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx Y Ty p : set, q = OrderedPair (OrderedPair (OrderedPair X Tx) (OrderedPair Y Ty)) p ∧ one_point_compactification X Tx Y Ty ∧ p ∈ Y ∧ ¬ p ∈ X ∧ (metrizable X Tx ↔ metrizable Y Ty)} of type set.
In Proofgold the corresponding term root is bc127f... and object id is b01255...
Definition. We define ex34_6_check_imbedding_proof to be {p ∈ 𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx f : set, p = OrderedPair (OrderedPair X Tx) f ∧ completely_regular_space X Tx ∧ Hausdorff_space X Tx ∧ embedding_of X Tx (power_real ω) (product_topology_full ω (const_family ω R)) f} of type set.
In Proofgold the corresponding term root is a13a70... and object id is 34202d...
Definition. We define ex34_7_locally_metrizable_compact_Hausdorff_metrizable to be {p ∈ 𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx : set, p = OrderedPair X Tx ∧ locally_metrizable_space X Tx ∧ compact_space X Tx ∧ Hausdorff_space X Tx ∧ metrizable X Tx} of type set.
In Proofgold the corresponding term root is 3d3899... and object id is 8219f8...
Definition. We define ex34_8_regular_Lindelof_locally_metrizable_metrizable to be {p ∈ 𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx : set, p = OrderedPair X Tx ∧ (regular_space X Tx ∧ Lindelof_space X Tx ∧ locally_metrizable_space X Tx → metrizable X Tx)} of type set.
In Proofgold the corresponding term root is b28321... and object id is 324edf...
Definition. We define ex34_9_compact_union_two_metrizable_closed_metrizable to be {p ∈ 𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx A B : set, p = OrderedPair (OrderedPair X Tx) (OrderedPair A B) ∧ compact_space X Tx ∧ Hausdorff_space X Tx ∧ closed_in X Tx A ∧ closed_in X Tx B ∧ ⋃ (UPair A B) = X ∧ metrizable A (subspace_topology X Tx A) ∧ metrizable B (subspace_topology X Tx B) ∧ metrizable X Tx} of type set.
In Proofgold the corresponding term root is b14394... and object id is 408bec...
Definition. We define ex35_1_Tietze_implies_Urysohn to be {p ∈ 𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx : set, p = OrderedPair X Tx ∧ normal_space X Tx ∧ (∀A B : set, closed_in X Tx A ∧ closed_in X Tx B ∧ A ∩ B = Empty → ∃f : set, continuous_map X Tx R R_standard_topology f ∧ (∀x : set, x ∈ A → apply_fun f x = 0) ∧ (∀x : set, x ∈ B → apply_fun f x = 1))} of type set.
In Proofgold the corresponding term root is 03f373... and object id is 0aa50f...
Definition. We define ex35_2_interval_partition_parameter to be {p ∈ 𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx : set, p = OrderedPair X Tx ∧ normal_space X Tx} of type set.
In Proofgold the corresponding term root is eebc23... and object id is d4a012...
Definition. We define ex35_3_boundedness_equivalences_metrizable to be {p ∈ 𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx d : set, p = OrderedPair (OrderedPair X Tx) d ∧ metric_on X d ∧ metric_topology X d = Tx} of type set.
In Proofgold the corresponding term root is 71984c... and object id is 8e738e...
Definition. We define ex35_4_retract_properties to be {p ∈ 𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx A : set, p = OrderedPair (OrderedPair X Tx) A ∧ retraction_of X Tx A} of type set.
In Proofgold the corresponding term root is d78f1f... and object id is 60cc11...
Definition. We define ex35_5_universal_extension_retracts to be {p ∈ 𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx A : set, p = OrderedPair (OrderedPair X Tx) A ∧ normal_space X Tx ∧ retraction_of X Tx A ∧ ∀Y Ty f : set, continuous_map A (subspace_topology X Tx A) Y Ty f → ∃g : set, continuous_map X Tx Y Ty g ∧ ∀x : set, x ∈ A → apply_fun g x = apply_fun f x} of type set.
In Proofgold the corresponding term root is 6e5542... and object id is c9306e...
Definition. We define ex35_6_absolute_retract_universal_extension to be {p ∈ 𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx : set, p = OrderedPair X Tx ∧ absolute_retract X Tx} of type set.
In Proofgold the corresponding term root is e776a5... and object id is 70006a...
Definition. We define ex35_7_retract_examples to be {p ∈ 𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx A : set, p = OrderedPair (OrderedPair X Tx) A ∧ retraction_of X Tx A} of type set.
In Proofgold the corresponding term root is d78f1f... and object id is 60cc11...
Definition. We define ex35_8_absolute_retract_equivalence to be {p ∈ 𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx : set, p = OrderedPair X Tx ∧ absolute_retract X Tx} of type set.
In Proofgold the corresponding term root is e776a5... and object id is 70006a...
Definition. We define ex35_9_coherent_topology_normal to be {p ∈ 𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx Y Ty : set, p = OrderedPair (OrderedPair X Tx) (OrderedPair Y Ty) ∧ (topology_on X Tx ∧ topology_on Y Ty ∧ coherent_topology X Tx Y Ty → normal_space Y Ty)} of type set.
In Proofgold the corresponding term root is 075364... and object id is 03a309...
Definition. We define ex36_manifold_embedding_exercises to be {p ∈ 𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃M TM f : set, p = OrderedPair (OrderedPair M TM) f ∧ m_manifold M TM → ∃n : set, embedding_of M TM (euclidean_space n) (euclidean_topology n) f} of type set.
In Proofgold the corresponding term root is b634d6... and object id is 529236...
Definition. We define ex37_tychonoff_exercises to be {p ∈ 𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃I Xi : set, p = OrderedPair I Xi ∧ compact_spaces_family I Xi ∧ compact_space (product_space I Xi) (product_topology_full I Xi)} of type set.
In Proofgold the corresponding term root is 3530fd... and object id is 57dc8d...
Definition. We define ex38_stone_cech_exercises to be {p ∈ 𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx Y Ty : set, p = OrderedPair (OrderedPair X Tx) (OrderedPair Y Ty) ∧ completely_regular_space X Tx ∧ compact_space Y Ty ∧ Hausdorff_space Y Ty ∧ ∃e : set, embedding_of X Tx Y Ty e} of type set.
In Proofgold the corresponding term root is 4b16fa... and object id is df723c...
Definition. We define ex39_local_finiteness_exercises to be {p ∈ 𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx U : set, p = OrderedPair (OrderedPair X Tx) U ∧ locally_finite_family X Tx U} of type set.
In Proofgold the corresponding term root is 83da58... and object id is f5b541...
Definition. We define ex40_nagata_smirnov_exercises to be {p ∈ 𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx B : set, p = OrderedPair (OrderedPair X Tx) B ∧ (regular_space X Tx ∧ basis_on X B ∧ locally_finite_family X Tx B → metrizable X Tx)} of type set.
In Proofgold the corresponding term root is 920e97... and object id is 0be928...
Definition. We define ex41_paracompactness_exercises to be {p ∈ 𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx U : set, p = OrderedPair (OrderedPair X Tx) U ∧ paracompact_space X Tx ∧ open_cover X Tx U} of type set.
In Proofgold the corresponding term root is 3440a8... and object id is 37ed1c...
Definition. We define ex42_smirnov_exercises to be {p ∈ 𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx B : set, p = OrderedPair (OrderedPair X Tx) B ∧ (regular_space X Tx ∧ basis_on X B ∧ locally_finite_family X Tx B → metrizable X Tx)} of type set.
In Proofgold the corresponding term root is 920e97... and object id is 0be928...
Definition. We define ex43_complete_metric_exercises to be {p ∈ 𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X d Tx : set, p = OrderedPair (OrderedPair X d) Tx ∧ metric_on X d ∧ Tx = metric_topology X d ∧ complete_metric_space X d} of type set.
In Proofgold the corresponding term root is 3c2422... and object id is 38964b...
Definition. We define ex44_space_filling_exercises to be {f ∈ 𝒫 (𝒫 (𝒫 R))|continuous_map unit_interval R2_standard_topology unit_square unit_square_topology f ∧ surjective_map unit_interval unit_square f} of type set.
In Proofgold the corresponding term root is 02abd1... and object id is 848149...
Definition. We define ex45_compact_metric_exercises to be {p ∈ 𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X d Tx : set, p = OrderedPair (OrderedPair X d) Tx ∧ metric_on X d ∧ Tx = metric_topology X d ∧ compact_space X Tx} of type set.
In Proofgold the corresponding term root is 72d77a... and object id is 9a0158...
Definition. We define ex46_convergence_exercises to be {p ∈ 𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx Y Ty : set, p = OrderedPair (OrderedPair X Tx) (OrderedPair Y Ty) ∧ topology_on X Tx ∧ topology_on Y Ty ∧ True} of type set.
In Proofgold the corresponding term root is 0eed3c... and object id is fde04a...
Definition. We define ex47_ascoli_exercises to be {p ∈ 𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx Y Ty : set, p = OrderedPair (OrderedPair X Tx) (OrderedPair Y Ty) ∧ compact_space X Tx ∧ Hausdorff_space Y Ty} of type set.
In Proofgold the corresponding term root is 30b8af... and object id is 979021...
Theorem. (ex48_1_Baire_union_interior)
∀X Tx : set, ∀Fam : set, Baire_space Tx → topology_on X Tx → X ≠ Empty → countable_set Fam → X = ⋃ Fam → ∃B : set, B ∈ Fam ∧ ∃U : set, U ∈ Tx ∧ U ≠ Empty ∧ U ⊆ (closure_of X Tx B)
In Proofgold the corresponding term root is 1c61f4... and proposition id is 9c8dc7...
Proof:
Let X, Tx and Fam be given.
Assume HBaire: Baire_space Tx.
Assume Htop: topology_on X Tx.
Assume Hnemp: X ≠ Empty.
Assume Hcount: countable_set Fam.
Assume Hunion: X = ⋃ Fam.
We will prove ∃B : set, B ∈ Fam ∧ ∃U : set, U ∈ Tx ∧ U ≠ Empty ∧ U ⊆ (closure_of X Tx B).
The rest of this subproof is missing.
∎
Theorem. (ex48_2_R_not_countable_empty_interior)
∀Fam : set, countable_set Fam → (∀C : set, C ∈ Fam → closed_in R R_standard_topology C ∧ (∀U : set, U ∈ R_standard_topology → U ⊆ C → U = Empty)) → R ≠ ⋃ Fam
In Proofgold the corresponding term root is a5fbed... and proposition id is 01a834...
Proof:
Let Fam be given.
Assume Hcount: countable_set Fam.
Assume Hnowhere: ∀C : set, C ∈ Fam → closed_in R R_standard_topology C ∧ (∀U : set, U ∈ R_standard_topology → U ⊆ C → U = Empty).
We will prove R ≠ ⋃ Fam.
The rest of this subproof is missing.
∎
Theorem. (ex48_3_locally_compact_Hausdorff_Baire)
∀X Tx : set, locally_compact X Tx → Hausdorff_space X Tx → Baire_space Tx
In Proofgold the corresponding term root is d0d7ea... and proposition id is f4e545...
Proof:
Let X and Tx be given.
Assume Hlc: locally_compact X Tx.
Assume HHaus: Hausdorff_space X Tx.
We will prove Baire_space Tx.
The rest of this subproof is missing.
∎
Theorem. (ex48_4_locally_Baire_implies_Baire)
∀X Tx : set, topology_on X Tx → (∀x : set, x ∈ X → ∃U : set, U ∈ Tx ∧ x ∈ U ∧ Baire_space (subspace_topology X Tx U)) → Baire_space Tx
In Proofgold the corresponding term root is efab67... and proposition id is 2f4534...
Proof:
Let X and Tx be given.
Assume Htop: topology_on X Tx.
Assume Hlocal: ∀x : set, x ∈ X → ∃U : set, U ∈ Tx ∧ x ∈ U ∧ Baire_space (subspace_topology X Tx U).
We will prove Baire_space Tx.
The rest of this subproof is missing.
∎
Theorem. (ex48_5_Gdelta_Baire)
∀X Tx Y : set, (compact_space X Tx ∧ Hausdorff_space X Tx) → (∃Fam : set, countable_set Fam ∧ (∀W : set, W ∈ Fam → W ∈ Tx) ∧ Y = intersection_over_family X Fam) → Baire_space (subspace_topology X Tx Y)
In Proofgold the corresponding term root is 36082c... and proposition id is e7bcb4...
Proof:
Let X, Tx and Y be given.
Assume Hcomp: compact_space X Tx ∧ Hausdorff_space X Tx.
Assume HGdelta: ∃Fam : set, countable_set Fam ∧ (∀W : set, W ∈ Fam → W ∈ Tx) ∧ Y = intersection_over_family X Fam.
We will prove Baire_space (subspace_topology X Tx Y).
The rest of this subproof is missing.
∎
Theorem. (ex48_6_irrationals_Baire)
Baire_space (subspace_topology R R_standard_topology (R ∖ Q))
In Proofgold the corresponding term root is cb7a25... and proposition id is 3a3a57...
Proof:
We will prove Baire_space (subspace_topology R R_standard_topology (R ∖ Q)).
The rest of this subproof is missing.
∎
Theorem. (ex48_7a_continuity_set_Gdelta)
∀f : set, function_on f R R → ∃Fam : set, countable_set Fam ∧ (∀U : set, U ∈ Fam → U ∈ R_standard_topology) ∧ {x ∈ R|continuous_at f x} = intersection_over_family R Fam
In Proofgold the corresponding term root is e30a09... and proposition id is a3e0df...
Proof:
Let f be given.
Assume Hf: function_on f R R.
We will prove ∃Fam : set, countable_set Fam ∧ (∀U : set, U ∈ Fam → U ∈ R_standard_topology) ∧ {x ∈ R|continuous_at f x} = intersection_over_family R Fam.
The rest of this subproof is missing.
∎
Theorem. (ex48_7b_countable_dense_not_Gdelta)
∀D : set, D ⊆ R → countable_set D → dense_in D R R_standard_topology → ¬ (∃Fam : set, countable_set Fam ∧ (∀W : set, W ∈ Fam → W ∈ R_standard_topology) ∧ D = intersection_over_family R Fam)
In Proofgold the corresponding term root is b5cbd1... and proposition id is 70d6fb...
Proof:
Let D be given.
Assume Hsub: D ⊆ R.
Assume Hcount: countable_set D.
Assume Hdense: dense_in D R R_standard_topology.
We will prove ¬ (∃Fam : set, countable_set Fam ∧ (∀W : set, W ∈ Fam → W ∈ R_standard_topology) ∧ D = intersection_over_family R Fam).
The rest of this subproof is missing.
∎
Theorem. (ex48_7_no_function_continuous_on_countable_dense)
∀D : set, D ⊆ R → countable_set D → dense_in D R R_standard_topology → ¬ (∃f : set, function_on f R R ∧ (∀x : set, x ∈ D → continuous_at f x) ∧ (∀x : set, x ∈ R → x ∉ D → ¬ continuous_at f x))
In Proofgold the corresponding term root is 393359... and proposition id is 9010e6...
Proof:
Let D be given.
Assume Hsub: D ⊆ R.
Assume Hcount: countable_set D.
Assume Hdense: dense_in D R R_standard_topology.
We will prove ¬ (∃f : set, function_on f R R ∧ (∀x : set, x ∈ D → continuous_at f x) ∧ (∀x : set, x ∈ R → x ∉ D → ¬ continuous_at f x)).
The rest of this subproof is missing.
∎
Theorem. (ex48_8_pointwise_limit_continuity)
∀fn : set, ∀f : set, (∀n : set, n ∈ ω → continuous_map R R_standard_topology R R_standard_topology (apply_fun fn n)) → function_on f R R → (∀x : set, x ∈ R → ∃limval : set, limval ∈ R ∧ ∀eps : set, eps ∈ R → True) → ¬ countable_set {x ∈ R|continuous_at f x}
In Proofgold the corresponding term root is fd82e0... and proposition id is baeb05...
Proof:
Let fn and f be given.
Assume Hfn: ∀n : set, n ∈ ω → continuous_map R R_standard_topology R R_standard_topology (apply_fun fn n).
Assume Hf: function_on f R R.
Assume Hlim: ∀x : set, x ∈ R → ∃limval : set, limval ∈ R ∧ ∀eps : set, eps ∈ R → True.
We will prove ¬ countable_set {x ∈ R|continuous_at f x}.
The rest of this subproof is missing.
∎
Theorem. (ex48_9_Thomae_function)
∀g : set, ∀f : set, (∀n : set, n ∈ ω → apply_fun g n ∈ Q) → function_on f R R → (∀n : set, n ∈ ω → apply_fun f (apply_fun g n) = R) → (∀x : set, x ∈ R → x ∉ Q → apply_fun f x = Empty) → ∀x : set, x ∈ R → x ∉ Q → continuous_at f x
In Proofgold the corresponding term root is 83890c... and proposition id is 0dda67...
Proof:
Let g and f be given.
Assume Hg: ∀n : set, n ∈ ω → apply_fun g n ∈ Q.
Assume Hf: function_on f R R.
Assume Hfg: ∀n : set, n ∈ ω → apply_fun f (apply_fun g n) = R.
Assume Hfirr: ∀x : set, x ∈ R → x ∉ Q → apply_fun f x = Empty.
Let x be given.
Assume Hx: x ∈ R.
Assume Hirr: x ∉ Q.
We will prove continuous_at f x.
The rest of this subproof is missing.
∎
Theorem. (ex48_10_uniform_boundedness)
∀X d : set, ∀FF : set, complete_metric_space X d → FF ⊆ 𝒫 (𝒫 R) → (∀a : set, a ∈ X → ∃M : set, M ∈ R ∧ ∀f : set, f ∈ FF → apply_fun f a ∈ R) → ∃U : set, ∃M : set, U ∈ (metric_topology X d) ∧ U ≠ Empty ∧ M ∈ R ∧ ∀f : set, f ∈ FF → ∀x : set, x ∈ U → apply_fun f x ∈ R
In Proofgold the corresponding term root is 4d538e... and proposition id is 40b726...
Proof:
Let X, d and FF be given.
Assume Hcomplete: complete_metric_space X d.
Assume HFF: FF ⊆ 𝒫 (𝒫 R).
Assume Hbound: ∀a : set, a ∈ X → ∃M : set, M ∈ R ∧ ∀f : set, f ∈ FF → apply_fun f a ∈ R.
We will prove ∃U : set, ∃M : set, U ∈ (metric_topology X d) ∧ U ≠ Empty ∧ M ∈ R ∧ ∀f : set, f ∈ FF → ∀x : set, x ∈ U → apply_fun f x ∈ R.
The rest of this subproof is missing.
∎
Theorem. (ex48_11_Rl_Baire)
∀Tl : set, Tl = R → Baire_space Tl
In Proofgold the corresponding term root is 54971e... and proposition id is d64b08...
Proof:
Let Tl be given.
Assume HTl: Tl = R.
We will prove Baire_space Tl.
The rest of this subproof is missing.
∎
Theorem. (ex49_1_verify_example_functions)
∀f g k : set, f = R → g = R → k = R → continuous_map R R_standard_topology R R_standard_topology f ∧ continuous_map R R_standard_topology R R_standard_topology g ∧ continuous_map R R_standard_topology R R_standard_topology k
In Proofgold the corresponding term root is 816340... and proposition id is 436767...
Proof:
Let f, g and k be given.
Assume Hf: f = R.
Assume Hg: g = R.
Assume Hk: k = R.
We will prove continuous_map R R_standard_topology R R_standard_topology f ∧ continuous_map R R_standard_topology R R_standard_topology g ∧ continuous_map R R_standard_topology R R_standard_topology k.
The rest of this subproof is missing.
∎
Theorem. (ex49_2_construct_bounded_function)
∀n : set, ∀eps : set, n ∈ ω → eps ∈ R → ∃f : set, continuous_map unit_interval R_standard_topology R R_standard_topology f ∧ (∀x : set, x ∈ unit_interval → apply_fun f x ∈ R) ∧ True
In Proofgold the corresponding term root is 5619e1... and proposition id is 5504f3...
Proof:
Let n and eps be given.
Assume Hn: n ∈ ω.
Assume Heps: eps ∈ R.
We will prove ∃f : set, continuous_map unit_interval R_standard_topology R R_standard_topology f ∧ (∀x : set, x ∈ unit_interval → apply_fun f x ∈ R) ∧ True.
The rest of this subproof is missing.
∎
Definition. We define ex50_dimension_exercises to be {p ∈ 𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx n : set, p = OrderedPair (OrderedPair X Tx) n ∧ topology_on X Tx ∧ ordinal n} of type set.
In Proofgold the corresponding term root is 4a8fe2... and object id is 5222cc...